I pp pi. GlU^l^ MATHEMATICAL MONOGRAPHS EDITED BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD No. 14 ALGEBRAIC INVARIANTS BY LEONARD EUGENE DICKSON Professor of Mathematics in the University of Chicago FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited 1914 Copyright, 1914, LEONARD EUGENE DICKSON V\^xw THE SCIENTIFIC PRESS ROBERT DRIMMOND AND COMPANY BROOKLYN, N. V. EDITORS' PREFACE. The volume called Higher Mathematics, the third edition of which was published in 1900, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume was discontinued in 1906, and the chapters have since been issued in separate Monographs, they being generally enlarged by additional articles or appendices which either amplify the former pres- entation or record recent advances. This plan of publication was arranged in order to meet the demand of teachers and the convenience of classes, and it was also thought that it would prove advantageous to readers in special lines of mathe- matical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the demand seems to warrant it. Among the topics which are under con- sideration are those of elliptic functions, the theory of quantics, the group theory, the calculus of variations, and non-Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this Series of Monographs may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. PREFACE This introduction to the classical theory of invariants of algebraic forms is divided into three parts of approximately equal length. Part I treats of linear transformations both from the stand- point of a change of the two points of reference or the triangle of reference used in the definition of the homogeneous coor- dinates of points in a line or plane, and also from the stand- point of projective geometry. Examples are given of invariants of forms / of low degrees in two or three variables, and the vanishing of an invariant of / is shown to give a geometrical property of the locus / = 0, which, on the one hand, is inde- pendent of the points of reference or triangle of reference, and, on the other hand, is unchanged by projection. Certain covariants such as Jacobians and Hessians are discussed and their algebraic and geometrical interpretations given; in particular, the use of the Hessian in the solution of a cubic equation and in the discussion of the points of inflexion of a plane cubic curve. In brief, beginning with ample illustra- tions from plane analytics, the reader is led by easy stages to the standpoint of hnear transformations, their invariants and interpretations, employed in analytic projective geometry and modern algebra. Part II treats of the algebraic properties of invariants and covariants, chiefly of binary forms: homogeneity, weight, annihilators, seminvariant leaders of covariants, law of reciproc- ity, fundamental systems, properties as functions of the roots, and production by means of differential operators. Any quartic equation is solved by reducing it to a canonical form by means of the Hessian (§33). Irrational invariants are illustrated by a carefully selected set of exercises (§ 35). V vi PREFACE Part III gives an introduction to the symbolic notation of Aronhold and Clebsch. The notation is first explained at length for a simple case; Hkewise the fundamental theorem on the types of symbolic factors of a term of a covariant of binary forms is first proved for a simple example by the method later used for the general theorem. In \dew of these and similar attentions to the needs of those making their first acquaintance with the symboHc notation, the difiiculties usually encountered will, it is beHeved, be largely avoided. This notation must be mastered by those who would go deeply into the theory of invariants and its apphcations. Hubert's theorem on the expression of the forms of a set linearly in terms of a finite number of forms of the set is proved and appHed to estabHsh the finiteness of a fundamental set of covariants of a system of binary forms. The theory of transvectants is developed as far as needed in the discussion of apolarity of binary forms and its appHcation to rational curves (§§ 53-57), and in the determination by induction of a fundamental system of covariants of a binary form without the aid of the more technical supplementary concepts employed by Gordan. Finally, there is a discussion of the t}'pes of s>Tn- boHc factors in any term of a concomitant of a system of forms in three or four variables, with remarks on line and plane coordinates. For further developments reference is made at appropriate places to the texts in EngUsh by Salmon, EUiott, and Grace and Young, as well as to Gordan's Invariantentheorie. The standard work on the geometrical side of invariants is Clebsch- Lindemann, Vorlesungen iiber Geometrie. Reference may be made to books by W. F. Meyer, Apolaritdt iind Rationale Curve, Bericht iiber den gegenwarligen Stand der Invariantentheorie, and FormentJieorie. Concerning invariant-factors, elementary divi- sors, and pairs of quadratic or bilinear forms, not treated here, see Muth, Elementartheiler , Bromwich, Quadratic Forms and their Classification by Means of Invariant Factors, and Bocher's Introduction to Higher Algebra. Lack of space prevents also the discussion of the invariants and covariants arising in the PREFACE vii theory of numbers; but an elementary exposition is available in the author's recent book, On Invaria?its and the Theory of Numbers, pubKshed, together with Osgood's lectures on func- tions of several complex variables, by the American Mathematical Society, as The Madison Colloquium. In addition to numerous illustrativ^e examples, there are four- teen sets of exercises which were carefully selected on the basis of experience with classes in this subject. The author is indebted to Professor H. S. White for suggest- ing certain additions to the initial list of topics and for reading the proofs of Part I. Chicago, May, 1914. TABLE OF CONTENTS PART I Illustrations, Geometrical Interpretations and Applications OF Invariants and Covariants PAGE § 1. Illustrations from Plane Analytics 1 § 2. Projective Transformations 4 § 3. Homogeneous Coordinates of a Point in a Line 8 § 4. Examples of Invariants 9 § 5. Examples of Covariants 11 § 6. Forms and Their Classification 14 § 7. Definition of Invariants and Covariants 14 Exercises 15 § 8. Invariants of Covariants 16 § 9. Canonical Form of a Binary Cubic: Solution of Cubic Equations 17 § 10. Covariants of Covariants 18 § 11. Intermediate Invariants and Covariants 19 Exercises 20 § 12. Homogeneous Coordinates of Points in a Plane 20 § 13. Properties of the Hessian 23 § 14. Inflexion Points and Invariants of a Cubic Curve 26 Exercises 28 PART II Theory of Invariants in Non-symbolic Notation §,15. Homogeneity of Invariants 30 § 16. Weight of an Invariant of a Binary Form 31 § 17. Weight of an Invariant of any System of Forms 32 Exercises 33 § 18. Products of Linear Transformations 33 § 19. Generators of all Binary Linear Transformations 34 § 20. Annihilator of an Invariant of a Binary Form 34 Example and Exercises ; 36 § 21. Homogeneity of Covariants 37 § 22. Weight of a Covariant of a Binary Form 38 § 23. Annihilators of Covariants 39 Exercises 40 X TABLE OF CONTENTS PAGE § 24. Alternants 41 § 25. Seminvariants as Leaders of Binary Covariants 42 § 26. Number of Linearly Independent Seminvariants 43 § 27. Hermite's Law of Reciprocity 45 Exercises 46 §§ 28-3L Fundamental System of Covariants 47 §§ 32, 33. Canonical Form of Binary Quartic; Solution of Quartic Equations. . 50 § 34. Seminvariants in Terms of the Roots 53 § 35. Invariants in Terms of the Roots 54 Exercises 55 § 36. Covariants in Terms of the Roots 56 Exercises 58 § 37. Covariant with a Given Leader 58 § 38. Differential Operators Producing Covariants 5!) Exercises 61 PART III Symbolic Notation §§ 39-41. The Notation and its Immediate Consequences 63 Exercises 65, 66 §§ 42-45. Covariants as Functions of Two Symbolic Types 67 § 46. Problem of Finiteness of Covariants 70 § 47. Reduction to Problem on Invariants 71 § 48. Hubert's Theorem on a Set of Forms 72 §§ 49, 50. Finiteness of a Fundamental System ot Invariants 73 § 51 . Finiteness of Syzygies 76 § 52. Transvectants 77 §§ 53, 54. Binary Forms Apolar to Given Forms 78 §§ 55, 56. Rational Plane Cubic Curves 81 § 57. Rational Space Quartic Curves 83 §§ 58, 59. Fundamental System of Covariants of Linear Forms; of a Quadratic Form ; Exercises 84 § 60. Theorems on Transvectants; Convolution 85 § 61. Irreducible Covariants Found by Induction 87 § 62. Fundamental System for a Binary Cubic 89 § 63. Results and References on Higher Binary Forms 91 § 64. Hermite's Law of Reciprocity Symbolically 91 § 65. Ternary Form in Symbolic Notation 92 Exercises ^^ §§ 66, 67. Concomitants of Ternary Forms 93 § 68. Quaternary Forms 97 Index 9^ ALGEBRAIC INVARIANTS PART I ILLUSTRATIONS, GEOMETRICAL INTERPRETATIONS AND APPLICATIONS OF INVARIANTS AND COVARIANTS. 1. Illustrations from Plane Analytics. If x and y are the coordinates of a point in a plane referred to rectangular axes, while x' and y' are the coordinates of the same point referred to axes obtained by rotating the former axes counter-clock- wise through an angle 6, then T: x = x' cos d — y' sin 6, ;y = a;' sin ^+7' cos ^. Substituting these values into the linear function l = ax-\-by-\-c, we get a'x' +b'y' -\-c, where a' =a cos d-\-h sin 6, h' = —a sin d+h cos 6. It follows that Accordingly, a?-\-lr is called an invariant of I under every transformation of the type T. Similarly, under the transformation T let L = Ax^By+C = A'x'-^B'y'-\-C, so that ^' = ^ cos ^+5 sin ^, B' = -Asm B+B cos 6. ALGEBRAIC INVARIANTS By the multiplication * of determinants, we get a' h' a h A' B' A B a' -h' a —b B' A' B A cos d —sin6 sin d cos d aB-bA, = aA-\-bB. cos 6 sm 6 -sin d cos 6 The expressions at the right are therefore invariants of the pair of linear functions I and L under every transformation of type T. The straight lines represented by 1 = and Z, = are parallel if and only if aB — bA=0; they are perpendicular if and only if aA-\-bB = 0. Moreover, the quotient of aB — bA by aA-\-bB is an invariant having an interpretation; it is the tangent of one of the angles between the two Hues. As in the first example, A^+B'^is an invariant of L. Between our four invariants of the pair /and L the following identity holds: {aA+bBy--{-{aB-bA)^ = (a^+b'~)(A^+B^-). The equation of any conic is of the form 5 = 0, where S = ax^-i-2bxy-\-cy^-\-2kx-\- 2ly + m . Under the transformation T, S becomes a function of x' and y\ in which the part of the second degree F = a'x'^+2b'x'y'+c'y'^ is derived solely from the part of S of the second degree: / = ax~ + 2bxy+cy". The coefficient a' of x'- is evidently obtained by replacing X by cos d and y by sin 6 in /, while c' is obtained by replacing X by — sin 6 and y. by cos d in /. It follows at once that a'-\-c' = a-{-c. Using also the value of b', we can show that a'c'-b'^ = ac-b^, * We shall always employ the rule which holds also for the multiplication of matrices: the element in the rth row and sth column of the product is found by multiplying the elements of the )-th row of the first determinant by the cor- responding elements of the 5th column of the second determinant, and adding the products. §1] ILLUSTRATIONS FROM ANALYTICS 3 but a more general fact will be obtained in § 4 without tedious multiplications. Thus a+c and d^ac — h- are invariants of /j and also of S, under every transformation of type T. When 5 = represents a real conic, not a pair of straight hues, the conic is an ellipse ii d>0, an hyperbola if (/<0, and a parabola if (^ = 0. When homogeneous coordinates are used, the classi- fications of conies is wholly different (§ 13). If X and y are the coordinates of a point referred to rectan- gular axes and if x' and ;-' are the coordinates of the same point referred to new axes through the new origin (r, s) and parallel to the former axes, respectivel}', then t: x = x'-\rr, y=y'-\-s. All of our former expressions which were invariant under the transformations T are also invariant under the new trans- formations t, since each letter a, b, . . . involved is invariant under t. But not all of our expressions are invariant under a larger set of transformations to be defined later. We shall now give an entirely different interpretation to the transformations T and /. Instead of considering {x, y) and {x\ y') to be the same point referred to different pairs of coordinate axes, we now regard them as different points referred to the same axes. In the case of /, this is accomplished by translating the new axes, and each point referred to them, in the direction from (r, s) to (0, 0) until those axes coincide with the initial axes. Thus any point {x, y) is translated to a new point {x' , y'), where x'=x—r, y'=y—s, both points being now referred to the same axes. Thus each point is translated through a distance Vr^-\-s^ and in a direction parallel to the directed hne from (0, 0) to {—r, —s). In the case of T, we rotate the new axes about the origin clockwise through angle 6 so that they now coincide with the initial axes. Then any point (x, y) is moved to a new point {x' , y') by a clockwise rotation about the origin through angle d. By solving the equations of T, we get x —X cos d-\-y sin d, y' = —x sin 0-f-y cos B. 4 ALGEBRAIC INVARIANTS These rigid motions (translations, rotations, and combinations of them) preserve angles and distances. But the transformation x' = 2x^ y' = 2y is a stretching in all directions from the origin in the ratio 2:1; while x' = 2x, y' = y is a stretching perpen- dicular to the ;y-axis in each direction in the ratio 2:1. From the multiplicity of possible types of transformations, we shall select as the basis of our theory of invariants the very restricted set of transformations which have an interpretation in projective geometry and which suffice for the ordinary needs of algebra. 2. Projective Transformations. All of the points on a straight line are said to form a range of points. Project the Fig. 1. points A, B, C, . . . oi a. range from a point V, not on their line, by means of a pencil of straight lines. This pencil is cut by a new transversal in a range ^i, Bi, Ci, . . . , said to be perspective with the range A, B, C, . . . . Project the points Ai, Bi, Ci, . . . from a new vertex v by a new pencil and cut it by a new transversal. The resulting range of points A', B' C, . . . is said to be projective with the range A, B, C, . . . Likewise, the range obtained by any number o.' projections and sections is called projective with the given range, and §2] PROJECTIVE TRANSFORMATIONS 5 the one-to-one correspondence thus estabhshed between cor- responding points of the two ranges is called a projec- tivity. To obtain an analytic property of a projectivity, we apply the sine proportion to two triangles in Fig. 1 and get 4C^sinj4FC ^C ^ sin ^FC AV'dnACV BV~smACV' From these and the formulas with D in place of C, we get AC^AV sin A VC AD^AT sin A VD BC BV' sin BVC BD BV' sin BVD' Hence, by division AC . AD _ sin A VC . sin A VD BC ' BD "sin BVC ' sin BVD' The left member is denoted by (A BCD) and is called the cross-ratio of the four points taken in this order. Since the right member depends only on the angles at V, it follows that {ABCD) = {AiBiCiDi), if .4i, . . . , Di are the intersections of the four rays by a second transversal. Hence if two ranges are projective, the cross-ratio of any four points of one range equals the cross- ratio of the corresponding points of the other range. Let each point of the line AB he determined by its dis- tance and direction from a fixed initial point of the line; let a be the resulting coordinate of A, and b, c, x those of B, C, D, respectively. Similarly, let A', B', C, D' have the coordinates a' , h' , c' , x' , referred to a fixed initial point on their line. Then (^5CZ))=— 4-^ = V47-V-w = (^'^'C'£>')• c— x — b c —b X —b Hence x' — b' _yX — b r,_c — a —, — K , k— r X —a x—a c-b ' c'-b' 6 ALGEBRAIC INVARIANTS SO that yfe is a finite constant ?^0, if C is distinct from A and B, and hence C distinct from A' and B'. Solving for x', we obtain a relation L: x' _«a;+/3 ^_ a /3 7 5 5^0. 7:r+5' In fact, ci = b - -ka\ ^ = ka'h-ah', 7 = 1- -^, 8 = bk-a If we multiply the elements of the first column of A by 5 and add the oroducts to the elements of the second column, we get A = b'-ka' b'{b-a) 1—k b—a = {b-a) -ka' b' -k 1 = k{b-a){b'-a')9^Q, if B and A are distinct, so that B' and A' are distinct. Hence a projectivity between two ranges defines a linear fractional transformation L between the coordinate x of a general point of one range and the coordinate x' of the corre- sponding point of the other range. The transformation is uniquely determined by the coordinates of three distinct points of one range and those of the corresponding points of the other range. If the ranges are on the same line and if A' = A, B'=^B, C'^C, then ^ = 1, a = 8, I3 = y = 0, and x' = x. Thus (ABCD) = {ABCD') implies D' = D. Conversely, if L is any given linear fractional transfor- mation (of determinant 5^ 0) and if each value of x is inter- preted as the coordinate of a point on any given straight line I and the value of x' determined by L as the coordinate of a corresponding point on any second given straight line /', the correspondence between the resulting two ranges is a pro- jectivity. This is proved as follows : Let A, B, C, D he the four' points of I whose respective coordinates are four distinct values Xi, X2, X3, X4: of x such that 'YXi-\-89^0. The corresponding values Xi, X2, xz, x^ of §2] PROJECTIVE TRANSFORMATIONS jc' determine four distinct points A', B\ C', D' of /'. For, if zV./, ,_ ,^ aXx + ^ aXj + _ A{Xi — Xj) yXi-\-8 yXj-{-8 {yXi+8){yXj+8) (A'B'C'D') = ''\~^\ ^ ''\~'^\ =r'^^^^^^^^^^ = (ABCD) Xs —X2 X4: —X2 X3 — X2 X4: — X2 since, if U denotes 7:^1 + 5, ^ ^ Wi/hkl ^ vwt/ w ^ ^' If A' 9^ A, project the points A', B' , C, D' from any con- venient vertex V' on to any line ABi through A and distinct Fig. 2. from I, obtaining the points Ai=A, B\, Ci, D\ of Fig. 2. Let V be the intersection of BBi with CC\ and let VD\ meet / at P. Then {ABCP) = (AiBiCiDi) = {A'B'CD') = (ABCD). From the first and last we have P = D, as proved above. Holding xi, X2, xz fixed, but allowing Xi to vary, we obtain two projective ranges on / and /'. If ^' = yl, we use T itself as yl^i and see that the ranges are perspective. 8 ALGEBRAIC INVARIANTS If / and /' are identical, we first project the range on /' on to a new line {A'B' in Fig. 2) and proceed as before. Any linear fractional transformation L is therefore a pro- jective transformation of the points of a line or of the points of one line into those of another line. The cross-ratio of any four points is invariant. 3. Homogeneous Coordinates of a Point in a Line. They are introduced partly for the sake of avoiding infinite coor- dinates. In fact, if 7 5^0, the value -8/y of x makes x' infinite. We set a; = a;i/a;2, thereby defining only the ratio of the homogeneous coordinates xi, X2 of a point. Leta[;'=:j;iVa[;2'. Then, if p is a factor of proportionaHty, L may be given the homogeneous form pxi =aXi-\-^X2, px2=yxi + 8x2, a8- ^y^O. The nature of homogeneous coordinates of points in a line is brought out more clearly by a more general definition. We employ two fixed points A and B of the line as points of reference. We define the homogeneous coordinates of a point P of the line to be any two numbers x, y such that x_ AP y'^'pB' where c is a constant 9^0, the same for all points P, while AP is a directed segment, so that AP=—PA. We agree to take y = if P^B. Given P, we have the ratio of x to y. Conversely, given the latter ratio, we have the ratio of ^P to PB, as well as their sum AP+PB = AB, and hence can find AP and therefore locate the point P. Just as we obtained in plane analytics {cf. § l) the relations between the coordinates of the same point referred to two pairs of axes, so here we desire the values of x and y expressed in terms of the coordinates ^ and rj of the same point P referred to new fixed points of reference A', B' . By definition, there is a certain new constant ^5^0 such that , "pB'- §3] HOMOGENEOUS COORDINATES 9 Since A'P-\-PB' =A'B\ we may replace A'P by A'B'-PB' and get p^,_ knA'B' Let A have the coordinates ^' , t] , referred to A\ B' . Then PA=PB'-AB' = PB'-^^ Jj'^-^:''}^-'^'^' Sirtiilarly, if B has the coordinates ^i, rji, referred to A' , B\ p^_ (v^i-^vi)k-A'B' Hence, by division, Since we are concerned only with the ratio of x to y, we may set Since the location of A and B with reference to A' and B^ is at our choice, as also the constant c (and hence r and s), the values of rr]' and —r^' are at our choice, likewise srjx and —s^\. There is, however, the restriction A 9^B, whence ij'^i j^ r)\ ^' Thus a change of reference points and constant multiplier c gives rise to a linear transformation X=a^-\-^T1, y^-Y^+brj, ^0, of coordinates, and conversely every such transformation can be interpreted as the formulas for a change of reference points and constant multiplier. 4. Examples of Invariants. The Unear functions l^ax-\-by, L=Ax-\-By become, under the preceding linear transformation T, where a' = aa-\-hy, b' = al3-\-b5, A' = Aa-\-By, B' = A^+BS. 10 ALGEBRAIC INVARIANTS Hence the resultant of the new linear functions is a' h' a b a iS = A a b A' B' A B 7 5 A B and equals the product of the resultant r — aB — bA of the given functions by A. Since this is true for every linear homogeneous transformation of determinant A, we call r an invariant of / and L of index unity, the factor which multiplies r being here the first power of A. Employing homogeneous coordinates for points on a line, we see that / vanishes at the single point {b,—a) and that L = only at {B,—A). These two points are identical if and only if b : a=B : A, i.e., if r = 0. The vanishing of the invariant r thus indicates a geometrical property which is independent of the choice of the points of reference used in defining coordinates on the line; moreover, the property is not changed by a projection of this line from an outside point and a section by a new line. Thus r = gives a projective property. Among the present transformations T are the very special transformations given at the beginning of § 1. Of the four functions there called invariants of / and L under those special transformations, r alone is invariant under all of the present transformations. Henceforth the term invariant will be used only when the property of invariance holds for all linear homo- geneous transformations of the variables considered. Our next example deals with the function / = ax^ + 2bxy + cy^ . The transformation T (end of § 3) replaces / by F=Ae+2B^v+Cv^ in which A =aa^-\-2bay-\-cy^, B = aa(3-\-b{a5-\-^j)-\-cy8, If the discriminant d = ac — b^oifis zero, / is the square of a linear function of x and v, so that the transformed function §4] EXAMPLES OF INVARIANTS 11 F is the square of a linear function of ^ and 77, whence the discriminant D = AC — B^ of F is zero. In other words, d = impHes D = 0. By inspection, the coefficient of —b-, the highest power of &, in the expansion of D is Thus D — A^d is a hnear function bq+r of b, where q and r are functions of a, c, a, /3, 7, 6. Let a sind_c remain arbitrary, but give to b the values Vac and —Vac in turn. Since d = and D = 0, we have 0=Vac^+r, 0=— Vac^+r, whence f = 9 = 0, D = A^d. Thus c/ is an invariant of / of index 2. Another proof is as follows: A^d = a 7 a b a /3 ^ 5 b c 7 5 a 7 aa-\-by 0/3+65 ^ B ^ 5 ba-{-c 7 ^'/J+c 5 B C = D. We just noted that d = expresses an algebraic property of /, that of being a perfect square. To give the related geo- metrical property, employ homogeneous coordinates for the points in a hne. Then /=0 represents two points which coin- cide if and only if d = 0. Thus the vanishing of the invariant d oi f expresses a projective property of the points represented by/=0. 5. Examples of Covariants. The Hessian (named after Otto Hesse) of a function /(x, y) of two variables is defined to be dj d'f h = dx'^ dxdy dydx dy^ Let /become F{^, 77) under the transformation T: x=a^-}-^v, y = 7^-f677, I \ = a 7 b 5^0. 12 ALGEBRAIC INVARIANTS Multiplying determinants according to the rule in § 1, we have hA = M dxdy + 5- dxdy a/' 3^93' dxdy 9/ dx dx dy dy where, by T, (1) ,=a^^,^=^^ + ^^^ = ^, dx dy 9a; 9^ 9y 9^ 9^ w- dx dy dv By the same rule of multiplication of determinants, / 9 , 9\9^ / d , d\dF \ dx dy/d^ \ dx dy/dv hA = dx dy) d^ \ dx dyj dv Applying (l) with/ replaced by dF/d^ for the first column and by dF/dn for the second column, we get 9^ _9^ de d~F A^h = drid^ 9^9r? 9^ 9r?2 Hence the Hessian of the transformed function F equals the product of the Hessian h of the given function / by the square of the determinant of the linear transformation. Conse- quently, h is called a covariant of index 2 of /. For an interpretation of /z = 0, see Exs. 4, 5, § 7. In case / is the quadratic function / of § 4, /f reduces to 'id, where d is the invariant ac — 62. The Junctional determinant or Jacohian (named after C. G. J. Jacobi) of two functions J{x, y) and g{x, y) is defined to be M df d{f,g)_ dx dy dix, y) dg dg dx dy EXAMPLES OF COVARIANTS 13 Let the above transformation T replace / by F{^, rj), and g by G(^, 7/). By means of (l), we get djF, G) d{^, v) 9/ , a/ dx dy dx dy M + 6^ dx dy dx dy dx dg dx dy dg dy I3\_^ d(f,g) 5 I d{x,yy Hence the Jacobian of / and g is a covariant of index unity of / and g. For example, the Jacobian of the linear functions / and Z, in § 4 is their resultant r; they are proportional if and only if the invariant r is zero. The last fact is an illus- tration of the Theorem. Two functions f and g of x and y are dependent if and only if their Jacobian is identically zero. First, U g = 4){f), the Jacobian of/ and g is 9/ dx 4>'{f) dx df_ dy = 0. Next, to prove the second or converse part of the theorem, let the Jacobian of / and g be identically zero. If g is a constant, it is a (constant) function of /. In the contrary case, the partial derivatives of g are not both identically zero. Let, for example, dg/dx be not zero identically. Consider g and y as new variables in place of x and >'. ThMs f=F{g,y) and the Jacobian is dFdg dFdg dg dx dg dy dg dg dx 3y dy ^ dF dy dx dy 14 ALGEBRAIC INVARIANTS Hence dF/dy is identically zero, so that F does not involve y explicitly and is a function of g only. 6. Forms and their Classification. A function like ax^-\-bx^y, every term of which is of the same total degree in x and y, is called homogeneous in x and y. A homogeneous rational integral function oi x, y, . . . is called a form (or quantic) m x, y, . . . . According as the number of variables is 1, 2, 3, . . . , or ^, the form is called unary, binary, ternary, . . . , or q-ary, respectively. Accord- , ing as the form is of the first, second, third, fourth, . . . , or p\h. order in the variables, it is called linear, quadratic, cubic, quartic, . . . , or p-ic, respectively. For the present we shall deal with binary forms. It is found to be advantageous to prefix binomial coefficients to the Hteral coefficients of the form, as in the binary quadratic and quartic forms ax^-{-2bxy-{-cy^, aox^-\-4:aix^y-^Qa2X^y^-}-4:a3xy^-\-a4y^. 7. Definition of Invariants and Covariants of Binary Forms. Let the general binary form / of order p, aox^+paix^-'y+^^^(^a2X^-y-^. . .-\-a,y^, be replaced by Ao^^+pAie-'v-\-^^~Y^A2e-'v'-\-- . .+^.^^ by the transformation T (§5) of determinant A?^0. If, for every such transformation, a polynomial I{ao, . . . , ap) has the property that 7(^0, . . . , ^p)=AV(ao, . . . , ap), identically in ao, • • • , «p, after the ^'s have been replaced by their values in terms of the a's, then I{ao, . . . , ap) is called an invariant of index X of the form/. §7] DEFINITION OF INVARIANTS 15 If, for every linear transformation T of determinant A?^0, a polynomial K in the coefficients and variables in / is such that * K{Aq, . . . ,Ay\ ^, r])^A^K(ao, . . . ,ap; X, y), identically in ao, . . . , ap, ^, -q, after the ^'s have been replaced by their values in terms of the a's, and after x and y have been replaced by their values in terms of ^ and rj from T, then K is called a covariant of index X of/. The definitions of invariants and covariants of several binary forms are similar. These definitions are illustrated by the examples in §§4, 5. Note that / itself is a covariant of index zero of /; also that invariants are covariants of order zero. EXERCISES 1. The Jacobian oi J=ax--\-2bxy+cy'^ and L = rx-\-sy is J = 2{as-br)x-\-2{bs-cr)y. If J is identically zero, f=tL'^, where t is a constant. How does this illustrate the last result in § 5? Next, let / be not identically zero. Let k and / be the values of x/y for which /=0; m that for which L = and n that for which 7 = 0. Prove that the cross-ratio (k, tn, I, n)= —1. Thus the points represented by/=0 are separated harmonically by those repre- sented by Z,=0, /=0. 2. If / is the Jacobian of two binary quadratic forms /and g, the points represented by 7 = separate harmonically those represented by /=0 and also those represented by g = 0. Thus 7 = represents the pair of double points of the involution defined by the pairs of points represented by/=Oandg = 0. 3. If /(x, y) is a binary form of order n, then (Euler) X — +y — = nf. dx ^dy ^ Hint : Prove this for /= ax^y" ~ ^ and for /= fi +f2. 4. The Hessian of (ax+by)"' is identically zero. Hint : It is sufficient to prove this for x^. Why? * The factor can be shown to be a power of A if it is merely assumed to be a function only of the coefficients of the transformation. 16 ALGEBRAIC INVARIANTS 5. Conversely, if the Hessian of a binary form/(x, y) of order n is iden- tically zero, / is the nXh. power of a linear function. Hints: The Hessian of / is the Jacobian of dj/ dx, dj/ dy- By the last result in § 5, these derivatives are dependent: cx , dy where a and b are constants. Solving this with Euler's relation in Ex. 3, we get {ax-^hy) -^ = naf, {ax+by) — = nbf, dx dy 9 log/_ na aJog/_ nb dx ax+by dy ax+by' Integrating, log;-;? log iax+by) = <i>iy)^^p{x). Hence (^ = 1// = constant, say log c. Thus f=c(ax + by)^. 8. Invariants of Covariants. The binary cubic form ( 1 ) jXx ,y)= ax^ + Zhx-y + 'icxy^ + dy^ has as a covariant of index 2 its Hessian 36 h: (2) h = rx^ + 2sxy + ty'^, r = ac-b~, 2s = ad-bc, t = bd-c^. Under any linear transformation of determinant A, let /become (3) F = A^^+-Wey}+^Cir^^+Dr^^. Let H denote the Hessian of F. Then the covariance of h gives (4) H = Re+'^S^r,-\-Tri^~=A^h, R = AC-B\ . . . Hence A^r, 2A~s, A-t are the coefficients of a binary quadratic form which our transformation replaces by one with the coeffi- cients R, 25, T. Since the discriminant of a binary quad- ratic form is an invariant of index 2, RT-S^ = AHA^r-AH-{A^sf\=A^{rt-s^). Hence rt — s^ is an invariant of index 6 of/. A Kke method of proof shows that any invariant of a covariant of a system of forms is an invariant of the forms. §9] CANONICAL FORM OF CUBIC 17 As an example in the use of the concepts invariants and covariants in demonstrations, we shall prove that the invariant * (5) - 4(r/ - s^) = {ad - hcf - 4(ac - h^) {hd - c^) is zero if and only \i J{x/y, l)=0 has a multiple root, i.e., if J{x, y) is divisible by the square of a linear function of x and y. If the latter be the case, we can transform / into a form (3) with the factor ^^] then C = D = and the function (5) written, in capitals is zero, so that the invariant (5) itself is zero. Conversely, if (5) is zero, /=0 has a multiple root. For, the Hessian (2) is then a perfect square and hence can be transformed into ^^, which, by the covariance of h, differs only by a constant factor from the Hessian R^- of the trans- formed cubic (3) . Thus6' = r = 0. IfZ) = 0, thenC = 0(byr = 0) and (3) has the factor ^^, as affirmed. If D^O, 9. Canonical Form of a Binary Cubic; Solution of Cubic Equations. We shall prove that every binary cubic form whose discriminant is not zero f can be transformed into X'^-\-Y^. For, if the discriminant (5) of the binary cubic (1) is not 2ero, the Hessian (2) is the product of two linear functions which are linearly independent. Hence the cubic form / can be transformed into a form F whose Hessian (4) reduces to 2S^r], and hence has i? = 0, T = Q, 5?^0. If C = 0, then 5 = (by i? = 0) and/^ = .4^3^Z)r7^^Z)?^0 {hy S^Q). Taking ^ = ^-^Y, r]=D-'Y, we get F = X^-\-Y^, as desired. The remaining case Cj^O is readily excluded; for, then Bp^O (by r = 0) and A=~, D = ^, AD = BC, 5 = 0. C B * It is often called the discriminant of/. It equals — a*P/27, where P is the product of the squares of the differences of the roots oif{x/y,\) =0. Other writers call a^P the discriminant of/. t If zero, / has a square factor and hence can be transformed into XW or X^. 18 ALGEBRAIC INVARIANTS To solve a cubic equation without a multiple root, we have merely to introduce as new variables the factors ^ and 7? of the Hessian. For, then, the new cubic is A |'^+Z)j?^ = 0. To treat an example, consider f=x^ + 6x-y + 12xy-+dy^ — 0. The Hes- sian is (d—8) {xy+2y^). Hence we take ^ = ^+23; and r] = y as new variables. We get /= ^' + (^—8)77^ If d = 9, we have ^^ + ri^ = 0, whence ^/n= —1, — w or —co^, where w is an imaginary cube root of unity. But x/y+2=^/r). Hence :K;/y=— 3, — w— 2, — co2-2. 10. Covariants of Covariants. Any covariant of a system of covariants of a system of forms is a covariant of the forms. The proof of this theorem is similar to that used in the following illustrations. We first show that the Jacobian of a binary cubic form / and its Hessian A is a covariant of index 3 of/. We have d(F,H) ^^ d(f,AVi) _^, d(f,h) a(^, v) d{x, y) d{x, y)' As the second illustration we consider the forms /, L in Ex. 1, § 7. Their Jacobian is the double of the covariant K = vx-\-wy of index unity, where v = as — br, w = hs — cr. Thus K and L are covariants of the system of forms/, L. These two linear covariants have as an invariant their resultant 1 = V w r s = as^ — 2brs-\-cr^. Under a linear transformation of determinant A, let / become A^^+. . . , and L become R^+Sr]. By the covariance of K, V^-{-Wv=A(.vx+wy), V=AS-BR, W = BS-CR. Thus our transformation replaces the hnear form having the coefficients A^; and Aw by one having the coefficients V and W. Th. resultant Av Aw r s §11] INTERMEDIATE INVARIANTS 19 of this linear form and L is an invariant of index unity. Hence V W R S AE, V W R S :A2 V w r s so that I = vs — wr is an invariant of index 2 of/ and L. From the earlier expression for /, we see that it is the resultant of / and L. We have therefore illustrated also the theorem that the resultant of any two binary forms is an invariant of those forms. 11. Intermediate Invariants and Covariants. From the invariant ac — b^ of the binary quadratic form f=ax^-\- 2hxy + cy~ we may derive an invariant of the system of forms / and /' where f = a'x^ + 2b'xy+c'y'^. Let any linear transformation replace / and / by If / is any constant, the form/+// is transformed into F+tF'. By the invariance of the discriminant of /+//', {A-VtA'){C+tC')-{B+tB'Y^6?\{a+ta'){cMc')-{b+tby\, identically in /. The equahty of the terms free of / states only the known fact that ac — b^ is an invariant of/. Similarly the equality of the terms involving t^ states merely that a'c' — b"^ is an invariant of/'. But from the terms multiplied by i, we see that (1) ac'^a'c-2bb' is an invariant of index 2 of the system of forms /, i' . It is said to be the invariant intermediate between their dis- criminants. It was discovered by Boole in 1841. The method is a general one. Let K be any covariant of a form jix.y, . . .). Let a, 6, ... be the coefhcients of /. Let fix, y, . . .) be a form of the same order with the coeffi- cients a' , b\ . . . . If in iT we replace a by a-\-ta', b by b+tb', . . . , and expand in powers of /, we obtain as the 20 ALGEBRAIC INVARIANTS coefficient of any power r of / a covariant of the system /, /'. By Taylor's theorem, this covariant is in which the symbolic rth power of 9/ 9a is to be replaced by d'/da', etc. EXERCISES 1. For r=l, K^ac — b\ (2) becomes (1). 2. Taking as K the Hessian (2) of cubic (1) in § 8, obtain the covariant {ac'+a'c-2bb')x'' + {ad'+a'd-bc'-b'c)xy + {bd' + b'd-2cc')y^ of index 2 of a pair of binary cubic forms. 3. If (1) is zero, the pair of points given by /=0 is harmonic with the pair given by/' = 0. 12. Homogeneous Coordinates of Points in a Plane. Let Li-. aiX+bty-hct = (f = l, 2, 3) be any three linear equations in x, y, such that ai h\ ci I A= a2 1)2 C2 5^0. ^3 &3 C2, I Interpret x and y as the Cartesian coordinates of a point referred to rectangular axes. Then the equations represent three straight lines Lt forming a triangle. Choose the sign before the radical in _aiPC-\-hiy_^tCi so that pi is positive for a point (x, y) inside the triangle and hence is the length of the perpendicular from that point to Li. The homogeneous (or trilinear) coordinates of a point {x, y) are three numbers Xi, X2, xs such that pXi=kipi, pX2=k2p2, pX3=k3p3, where ki, k2, ks are constants, the same for all points. In view of the undetermined common factor p, only the ratios of xi, X2, X3 are defined. §12] HOMOGENEOUS COORDINATES 21 For example, let the triangle be an equilateral one with sides of length 2, base on the .r-axis and vertex on the y-axis. The equations of the sides Li, Li, Lz are, respectively, Vz+'='' vr'='- "''■ Take each ^<=1. Then y + V3(x-l) y-Vz{x + \) pxi= ;; , pXi= , pX3=y. -2 _2 The curve XiX^^Xi"^ is evidently tangent to Z,i(i.e., Xi = 0) at Q=(010), and tangent to Li at P=(100). Substituting for the x< their values, we see that the Cartesian equation of the curve is ix-o Fig. 3. i{(>'-V3)^-3x=}=y2orx2+(^y+^y-|. Hence it is a circle with radius CP and center at the intersection C of the normal to L^ at P with the normal to Zi at Q. Changing the notation for the coeflScients of kipi, call them a J, hi, d. Then we have m pXi = aiX-{-biy-\-Ci, A?^0 (^• = 1,2,3). 22 ALGEBRAIC INVARIANTS Multiply the ith equation by the cof actor Ai of at in the determinant A and sum for i = l, 2, 3. Next use as multiplier the cofactor Bt of bi] finally, the cofactor d of d. We get Ax = p'EAiXi, Ay = pi:BiXi, A = p^CtXi. Hence x and y are rational functions of x\, X2, xs: , „. _ AiXi-{-A2X2-{-A3X3 ^ BiXi-\-B2X 2-i-B3X3 CiXi-\-C2X2 + CsX3' CiXi-\-C2X2-hC3Xs' Any equation f{x, y)=0 in Cartesian coordinates becomes, by use of (C), a homogeneous equation (j)(xi, X2, X3)=0 in homogeneous coordinates. The reverse process is effected by use of (H). In particular, since any straight line is represented by an equation of the first degree in x and y, it is also rep- resented by a homogeneous equation of the first degree in Xi, X2, X3. For example, the sides of the triangle of reference are xi = 0, X2 = 0, X3 = 0. Conversely, any homogeneous equation of the first degree in xi, X2, X3 represents a straight line. The degree of ^ is always that of /. Take the y-axis as Li, the x-axis as L2, and let L3 recede to infinity by making C3 and 63 approach zero. Then (//) and (C) become Xi Xi pXi = x, pxi.—y, pxz=\; x=-, y=~. X3 X3 We are thus led to a very special, but much used, method of passing from homogeneous to Cartesian coordinates and conversely. For a new triangle of reference, let the homogeneous coor- dinates of (jc, y) be yi, y2, ys- Then, as in (5'), pyi = a'iX^rh'iy^rc'i (i = l,2, 3). Inserting the values of x and y from (C), we get relations like i'. ryi = eax -\-f1X2 +g<X3 (^ = 1 , 2, 3) . Hence a change of triangle of reference and constants ki, k2, ks gives rise to a linear homogeneous transformation / of coordinates. The determinant of the coefficients in / is not §13] PROPERTIES OF THE HESSIAN 23 zero, since yi=0, >'2 = 0, y3 = represent the sides of the new triangle. Conversely, any such transformation t may be interpreted as a change of triangle of reference and con- stants ki. Instead of regarding t as a set of relations between the coordinates of the same point referred to two triangles of reference, we may regard it as defining a correspondence between the points {xi, X2, X3) and (yi, y2, ys) of two different planes, each referred to any chosen triangle of reference in its plane. This correspondence is projective; for, it can be effected by a series of projections and sections, each projection being that of the points of a plane from a point outside of the plane and each section being the cutting of such a bundle of pro- jecting lines by a new plane. Proof will not be given here, nor is the theorem assumed in what follows. It is stated here to show that if / is any invariant of a ternary form / under all linear transformations /, then 7 = gives a projective property of the curve /=C. It is true conversely that any projective transformation between two planes can be effected by a linear homogeneous transformation on the homogeneous coordinates. ' Thus for three variables, just as for two (§§ 2, 3), the investigation of the invariants of a form under all linear homogeneous transformations is of e^i^ecial imoortance. 13. Properties of the Hessian. Let f{xi, . . . , Xn) be a form in the independent variables xi, . . . , Xn- The Hessian /j of / is a determinant of order w in which the elements of the ith. row are a-/ dj ay dXidxi dXtdX2 dxtdXn Let /become <i){yi, . . . , jn) under the transformation T: Xi = Ciiyi+Ci2y2-\-. . .+Ci„y„ (/ = !,. . . , w), of determinant A = |q|. The product hA is a determinant of order n in which the element in the ith row and jth column is the sum of the products of the above elements of the ^'th 24 ALGEBRAIC INVARIANTS row of h by the corresponding elements of the ^th column of A, and hence is -C\}-\-- — r C2j-\-. . .+— — Cn) ^ a / 9/ dxi df dX2 . ■ df dxn\ ^ d dxXdxidyj dX2dyj ' ' ' dxndyjj dx d± dXidyj' = Hessian of 0. = 1 n Let a' be the determinant obtained from A by interchanging its rows and columns. In the product A'-M, the element in the rth row and jih column is therefore dxi dyj dXn dyj dyr dyj since dr is the partial derivative of Xt with respect to yr. Hence dyrdyj Thus h is Si covariant of index 2 of/. To make an application to conies, let / be a ternary quad- ratic form. Then h is an invariant called the discriminant of/. Let (ai, 02, as) be a point on /=0 (for example, one with 0:3 = 0). For Ca=ai and Ct2, Cjs chosen so that A?^0, transformation T makes (x) = (a) correspond to (3;) = (100). Hence we may assume that (100) is a point on/=0, so that the term in xi^ is lacking. Consider the terms xj with the factor xi. If /^O, / involves only X2 and xs and hence is a product of two linear functions, while h = 0. In the contrary case, we may introduce / as a new variable in place of 0:2. This amounts to setting l = X2, f = XiX2-\-aX2^-\-bX2X3-\-CX3^. Replacing xi by Xi — ax2 — bx3, we get :j;ia;2 — ^^3^, whose Hessian is 2k. Hence /=0 represents two (distinct or coincident) straight lines if and only if the Hessian (discriminant) of / is zero. Moreover, if the discriminant is not zero, then k^^O and we may replace Vkxs by X3 and get 0:1^:2— ^3^- Hence all conies, which do not degenerate into straight lines, are equivalent § 13] PROPERTIES OF THE HESSIAN 25 under projective transformation. If the triangle of reference is equilateral and the coordinates are proportional to the per- pendiculars upon its sides, :riX2— X3^ = is a circle (§ 12). On the contrary, if we employ only translations and rota- tions, as in plane analytics, there are infinitely many non- equivalent conies ; we saw in § 1 that there are then two invariants besides the discriminant. Next, to make an application to plane cubic curves, let f{xx, X2, xz) be a ternary cubic form. A triangle of reference can be chosen so that P=(001) is a point of the curve /=0. Then the term in x^^ is lacking, so that / = ^3-/l+A-3/2+/3, where /« is a homogeneous function of .ri and X2 of degree /. We assume that P is not a singular point, so that the partial derivatives of / with respect to Xi, X2, and xa are not all zero at P. Hence /i is not identically zero and can be introduced as a new variable in place of xi. Thus, after a preliminary linear transformation, we have xz^xi+xz{axi~ -[-hxiX2+cx2^) +fi. Replace xs by xs — ^ (axi -\-bX'2) ■ We get F = xs^xi -\-ex3X2-+C, where C is a cubic function of Xi, X2, whose second partial derivative with respect to Xt and Xj will be denoted by Cy. The Hessian of F is H = If the transformation which replaced / by F is of deter- minant A, it replaces the Hessian k of / by H = A~h. Thus H = represents the same curve as h = 0, but referred to the same new triangle of reference as F = 0. We may therefore speak of a definite Hessian curve of the given curve f=0. In investigating the properties of these curves we may therefore Cn Ci2 2X3 Cv2 C22-\-2ex3 2eX2 2X3 2ex2 2xi 26 ALGEBRAIC INVARIANTS refer them to the triangle of reference for which their equations aren = 0,F = 0. The coefficient of xs^ in H is evidently — 8^. Thus P is on the Hessian curve if and only ii e = 0. li d is the coefficient of X2^ in C,xi=0 meets 7^ = at the points for which X2"{ex3-\-dx2) =0 and these points coincide (at F) if and only if e = 0. In that case, P is called a point of inflexion oi F = and a-i=0 the inflexion tangent at P. For a cubic curve /=0 witJwut a singular point, every point of inflexion is a point of intersection of the curve with its Hessian curve and conversely. 14. Inflexion Points and Invariants of a Cubic Curve. EHm- inating x^ between / = 0, A = 0, we obtain a homogeneous relation in Xi, X2, which has therefore at least one set of solutions .t'i, X2- For the latter values of Xi and X2, /=0 and h = are cubic equations in xs with at least one common root, x's. Hence /=0 has at least one inflexion point {x'l, x'2, x'3). After a suitable linear transformation, this point becomes (001). As in § 13, we can transform / into F, in which e is now zero. If d = 0, then F = xiQ, and the derivatives =(J-rXi , Xi , — — —^i~ dxi 9x1 9X2 9x2 9x3 9x3 all vanish at an intersection of xi=0,Q = 0. But we assume that there is no singular point on/=0 and thus none on F=0. Hence J 5^0. Replacing X2 by J~*X2, we have an F with d = l. Adding a multiple of Xi to X2, we get F=X32xi+C, C=X2=^+3iX2Xi2+aXi3, di Ci C12 C2 ^=-4X32C22 + 2Xi</., <, so that is the Hessian of C. By § 8, (f, = 3Q{ — b-xr-{-axiX2+bx2^). Eliminating xs^ between F = 0, ^^ = 0, we get Xi20 + 2C22C= 12(x2* + 6^)X22xi2+4ax2Xi3-362xi4) =0. If xi = 0, then X2 = and we obtain the known intersection § 14] INFLEXION POINTS OF CUBIC 27 (001). For the remaining intersections, we may set Xi = l and obtain from eacii root r of (1) r4+6&/^+4ar- 3^2 = two intersections (1, r, ix's). For, if x'z = ^, then C = 0, so that (1) would have a multiple root, whence d?'-\-W' = ^. But the three partial derivatives of F would then all vanish at {2b, ~a, 0) or (1,0,0), according as ftj^O or b = 0. Hence there are exactly nine distinct points of inflexion. For each of the four roots of (1), the three points of inflexion P and (1, r, ztx's) are collinear, being on X2=rxi. Since we may proceed with any point of inflexion as we did with P, we see that there are 9-4/3 or 12 lines each joining three points of inflexion and such that four of the lines pass through any one of the nine points. The six points of inflexion not on a fixed one of these lines therefore lie by threes on two new lines; three such lines form an inflexion triangle. Thus there are ^12 = 4 inflexion triangles. The fact that there are four inflexion triangles, one for each root r of (1), can also be seen as follows: iTH+rF = {rxi-X2)\x3~-rx2^-{r^-\-3b)xiX2-{r^-{-Qbr-\-3a)xi^\. The last factor equals X3'--\rx2+hir^-\-3b)xir-, r and hence is the product of two hnear functions. Corresponding results hold for any cubic curve /=0 without singular points. We have shown that / can be reduced to the special form 7^ by a linear transformation of a certain determinant A. Follow this by the transformation which multiplies xs by A and Xi by A"^^ and hence has the determin- ant A~^ Thus there is a transformation of determinant unity which replaces / by a form of type F, and hence replaces the Hessian h oi f by the Hessian H of F. Hence there are exactly four values of r for which i}/ = h-\-2Arf has a linear factor and therefore three linear factors. These r's are the roots of a quartic (l) in which a and b are functions of the coefi&cients 28 ALGEBRAIC INVARIANTS of/. To see the nature of these functions, let Xi—\X2—nX3 be a factor of \p. After replacing xi by 'Kx2+fJiX3 in \l/, we obtain a cubic function of X2 and X3 whose four coefficients must be zero. Eliminating X and ai, we obtain two conditions involving r and the coefficients of / rationally and integrally. The greatest common divisor of their left members is the required quartic function of r. Unless the coefficient of r^ is constant, a root would be infinite for certain /'s. The inflexion triangles of a general cubic curve /=0 are given by h-{-2^rf=0, where h is the Hessian of f and r is a root of the quartic (1) in which a and b are rational integral invariants of f. The explicit expressions for these invariants are very long; they are given in Salmon's Higher Plane Curves, §§ 221-2, and were first computed by Aronhold. For their short sym- bolic expressions, see § 65, Ex. 4. EXERCISES 1. Using the above inflexion triangle yiy-iys — O, where rxi — X2 = yY, ^rXi±:{rx-i-\-kxx) = 2yi, 2yi, k={r^+2.b)/2, r^+k = l{r'- + h)9^Q, as shown by use of (1), we have the transformation ^rXi = yn-\-yi, {r--\-k)xi = ryi+D, {r'^+Ji)x2= —kyi-\-rD, where D = y2—y:u Using (1) to eliminate a, show that ^ {r".j^b)F = -{y-^-y,') +Zyxyty,--(r^+U)yiK 8 r 8 Adding the product of the latter by 54 to its Hessian, we get the product of yiyiyz by Z\r'^ +h) / r'^ . Hence the nine points of inflexion are found by setting yi, y^, yz equal to zero in turn. 2. By multiplying the y's in Ex. 1 by constants, derive /= a(3,3 +s,3 +233) +6|3ZiZ2Z3, called the canonical form. Its Hessian is ^%, where // = -aiS^Si' +S..' +23=') + (a3 +2/33)ZiZ223. Thus find the nine inflexion points and show that the four inflexion triangles are ZiZ2Z3 = 0, 2Zi'— 3/2iZiZ3 = (/=!, w, w^), 14] INFLEXION POINTS OF CUBIC 29 where w is an imaginary cube root of unity. Their left members are constant multiples of 3h+rf, where r = 3/32, — (/«— ^)2 are the four roots of (1), with 3. The Jacobian of /i(a;i, . . . , Xn), . . . , fnixi, . . , Xn) is dfndfn Show that it is a covariant of index unity of /i, . . . , fn- 4. Hence the resultant of three ternary linear forms is an invariant of index unity. 5. If /i, . . . ,fn are dependent functions, the Jacobian is zero. PART II THEORY OF INVARIANTS IN NON-SYMBOLIC NOTATION 15. Homogeneity of Invariants. We saw in § 11 that two binary quadratic forms / and /' have the invariants d = ac-b'^, s = ac'+a'c-2bh' of index 2. Note that s is of the first degree in the coefficients a, b, c oi f and also of the first degree in the coefficients of /', and hence is homogeneous in the coefficients of each form separately. The latter is also true of d, but not of the invariant s-\-2d. When an invariant of two or more forms is not homogeneous in the coefficients of each form separately, it is a sum of invariants each homogeneous in the coefficients of each form separately. A proof may be made similar to that used in the following case. Grant merely that s-\-2d is an invariant of index 2 of the binary quadratic forms/ and/'. In the transformed forms (§ 11), the coefficients A, B, C oi F are linear in a, b, c; the coefficients A', B' , C of F' are Hnear in a', b', c' . By hypothesis AC ^A'C-2BB' ^2{AC-B'^)=^\s^2d). The terms 2d!^ of degree 2 in a, b, c on the right arise only from the part 2(AC-B^) on the left. Hence d is itself an invariant of index 2; likewise s itself is an invariant. However, an invariant of a single form is always homo- geneous. For example, this is the case with the above dis- criminant d of /. We shall deduce this theorem from a more general one. 30 §15] HOMOGENEITY OF INVARIANTS 31 Let / be an invariant of r forms /i, . . . ,/r of orders p\y . . ., pT in the same q variables Xi, . . . , Xq. Let a particular term ^ of / be of degree di in the coefficients of /i, of degree d2 in the coefficients of /2, etc. Apply the special transformation Xi=a^i, X2=(x^2, • • ., XQ=a^g, of determinant A =a'^. Then/t is transformed into a form whose coefficients are the products of those of ft by a^K Hence in the function / of the transformed coefficients, the term cor- responding to / equals the product of t by This factor therefore equals A^, if X is the index of the invariant. Thus r S dipi = \q. Hence I,dipi is constant for all the terms of the invariant. For the above two quadratic forms, r = pi — pi = 2. For invariant d, we have di = 2, di = 0, '^dipi = 4: = 2\. For s, we have di = d.,= l, Zdtpi = 4:. Again, the discriminant (§8) of the binary cubic form is of constant degree 4 and index X = 6; we have 7:dipi = i-3 = 2\. If, as in the last example, we take r = l, we see that an invariant of index X of a single ^-ary form of order p is of constant degree d, where dp = X^, and hence is homogeneous. 16. Weight of an Invariant / of a Binary Form f. Give to I and/ the notations in § 7. Let be any term of /, and call ^ = ^1+2^2+3^3 + . . .+pep the weight of /. Thus w is the sum of the subscripts of the factors Oi each repeated as often as its exponent indicates. We shall prove that the various terms of an invariant of a binary form are of constant weight, and hence call the invariant isoharic. For example, aQX^-\-2a\xy-\-a2y^ has the invariant 00^2— fli^, each of whose terms is of weight 2. 32 ALGEBRAIC INVARIANTS To prove the theorem, apply to/ the transformation X=^, y=ari. We obtain a form with the Uteral coefficients Ao = ao, Ai=aia, A2 = a2a^, . . . , Ap = apa^. Hence if I is of index X, /(go, aia, . . . , apaP)=a^I{ao, ai, . . . , Up), identically in a and the a's. The term of the left member which corresponds to the above term / of / is evidently CoQo^o . . . Qp^'va^. Hence w = X. The weight of an invariant of degree d oi 2. binary p-\z is thus its index and hence (§ 15) equals \dp. 17. Weight of an Invariant of any System of Forms. Let /i, . . . , /„ be forms in the same variables X\^ . . . , Xg. We define the weight of the coefficient of any term of ft to be the exponent of Xa in that term, and the weight of a product of coefficients to be the sum of the weights of the factors. For q = 2, this definition is in accord with that in § 16, where the coefficient at of a:iP~%2^' was taken to be of weight k. Again, in a ternary quadratic form, the coefficients of xi^, a;iit;2 and X2^ are of weight zero, those of X1X3 and ::C2^3 of weight unity, and that of xs^ of weight 2. Under the transformation of determinant a, Xl = §1, . . . , Xq —I =^ ^q—l, Xg =Q;^j, fi becomes a form in which the coefficient c' corresponding to a coefficient c of weight k in ft is ca/^. If / is an invariant, 7(c')^q:V(c), identically in a. Hence every term of I is of weight X. Thus any invariant of a single form is isoharic; any invariant of a system of two or more forms is isoharic on the whole, hut not necessarily isoharic in the coefficients of each form separately. The index equals the weight and is therefore an integer ^ 0. §18] PRODUCTS OF LINEAR TRANSFORMATIONS 33 EXERCISES 1. The invariant a^a' 2+a2a' Q—2aia' i of Gox"^ -\-2aixy+a2y~, a' ox"^ -\-2a\xy -\-a'2y'^ is of total weight 2, but is not of constant weight in ao, Ci, a2 alone. 2. Verify the theorem for the Jacobian of two binary h'near forms. 3. Verify the theorem for the Hessian of a ternary quadratic form. 4. No binary form of odd order p has an invariant of odd degree d. ^0, 18. Products of Linear Transformations. The product TT' of a 13 7 5 ' a' a p 7 is defined to be the transformation whose equations are obtained by eliminating ^ and r? between the equations of the given transformations. Hence la"=aa'+/37',i8"=a/3'+/35',7" = 7a' + 57',5" = 7/3' + 55'. Its determinant is seen to equal AA' and hence is not zero. By solving the equations which define T, we get ,.5 /3 —7 a A A-^ A A-^ These equations define the transformation T~^ inverse to T\ each of the products TJ"^ and T~'^T is the identity trans- formation a; = X, y=Y . The product of transformation Tq, defined in § 1, by T^' is seen to equal Tgj^Q', in accord with the interpretation given there. The inverse of Tb is T -e: ^ = X cos e -\-y ?,\n e , 77= — x sin e+y cos 0. Consider also any third linear transformation Ti: X=axU+&iV, Y = yiU+biV. To prove that the associative law {Tr)Ti = T{T'Ti) 34 ALGEBRAIC INVARIANTS holds, note that the first product is found by eliminating first ^, rj and then X, Y between the equations for T, T, Ti, while the second product is obtained by eliminating first A"^, Y and then ^, 7] between the same equations. Thus the final eliminants must be the same in the two cases. Hence we may write TT'Ti for either product. 19. Generators of All Binary Linear Transformations. Every binary linear homogeneous transformation is a product of the transformations Tn: x=^+nri, y = v; S,: x=^, y = kr) (k^O); V: x=-v, y=^- From these we obtain * ]/'-i = F^: x = v, y= — ^', V-^T-nV=^T'n: x = x', y = y-\-nx'; V-'StV =S\: x = kx', y-^y' (k^O). For 55^0, the transformation T in § 18 equals the product For 5 = 0, SO that iSy^^O, T equals SyS' -pi -a/pV ' 20. Annihilator of an Invariant of a Binary Form. The binary form in § 7 may be written as either of the sums /=S y.\ aiX^-^y = ^ y.\ ap-iX'yP-K Transformation V, of determinant unity, replaces the second sum by ii^^a,-t{-ly^r>-^r,i, Comparing this with the first sum we see that an invariant of /must be unaltered when (1) di is replaced by { — lYap-i (i = 0, 1, . . . ,p), * The T's are of the nature of translations, and the 5's stretchings. § 20] ANNIHILATOR OF INVARIANTS 35 By § 16, a function I{ao, ..., Gp) is invariant with respect to every transformation St if and only if it is isobaric. Finally, the function must be invariant with respect to every r»; under this transformation let Differentiating partially with respect to n, we get since T7 = y is free of n, while ^ = x — nri. The total coeflScient of ^p-V is the second term being absent if 7=0. But Hence] dn dAi dAo QA3 ^ QAp Now I(ao, . . . , dp) is invariant with respect to every transformation T„, of determinant unity, if and only if I{Ao, . . . , Ap)=I{aQ, . . . , dp), identically in n and the a's. This relation evidently implies dljAo, • • • , ^p) ^Q Conversely, the latter implies that /(^o, . • . , Ap) has the same value for all values of n and hence its value is that given by n = 0, viz., I(ao, . . . , Gp). Hence / has the desired property if and only if the right member of (2) is zero identically in n and the a's. But this is the case if and only if f2/(ao, . . . , ap)=0, 36 ALGEBRAIC INVARIANTS identically in the a's, where 2 is the differential operator dai 3^2 das 9«p In other words, I must satisfy the partial differential equation 12/ = 0. In Sylvester's phraseology, / must be anni- hilated by the operator Q. From this section and the preceding we have the important Theorem. A rational integral function I of the coefficients of the binary form f is an invariant of f if and only if I is iso- baric, is unaltered by the replacement (1), and is annihilated by n. EXAMPLE An invariant of degree d of the binary quartic (§6) is of weight 2d (end of § 16). For J=l, the only possible term is kai] since 0=U{ka2) = 2kai, we have ^=0, For J =2, we have / = raodi +saia3 +ta2'^, i2/= (5+4r)aoa3 + (4/+35)aia2 = 0, 5=— 4r, i = 3r, / = r(aoa«-4aia3+3cj*). EXERCISES 1. Every invariant of degree 3 of the binary quartic is the product of a constant by J = ao(i2ai-\-2aia2a3 — Goaz'^ — Oi^ai— a2^. 2. The invariant of lowest degree of the binary cubic aoX^+3aiX^y+3a2xy'^+a3y^ is its discriminant (aoOs— oiOa)^— 4(ao02— ai^)(aia3— 02^). 3. An invariant of two or more binary forms C0XP1+. . ., boxP^+. . ., CoxP^+. . . is annihilated by the operator sn^oo— +2a,— +. . .+bo~-+2bi~+. . .+Co— +. . . . dai da2 dbi dbz dCi 4. Every invariant of aox^+2aixy+a2y^, boX^+2bixy+b2y^ §21] HOMOGENEITY OF COVARIANTS 37 of the first degree in the a's and first degree in the 6's is a multiple of a<,bi+a2bo—'2aibi. 5. A binary quadratic and quartic have no such lineo-hnear invariant. 6. Find the invariant of partial degrees 2, 1 of a binary linear and a quadratic form. 7. Find the invariant of partial degrees 1, 2 of a binary quadratic and a cubic form. 8. The first two properties in the theorem of § 20 imply that / is homo- geneous. For, under replacement (1), any term cao% . . . Up^P of /, of weight w = ei+2e2+ . . . +pep, implies a term zLcao^^ai^P-i . . . ap% of weight w~ep-i+2ep-2+ . . . + ip — l)ei+ peo. Adding the two expressions for w, show that the degree d = ea+ei+ . . . -{-Cp is the constant 2w/p. 21. Homogeneity of Covariants. A covariant which is not ho}}ws;cncous in the variables is a sum of covariants each homo- geneous in the variables. For, il a, b, . . . are the coefficients of the forms, and K is a covariant, K(A, B,. . .; ^,v, ■ • .)=A^K(a, b, . . .; x, y, . . .). When X, y, . . . are replaced by their linear expressions in I, 77, ... , the terms of order co in x, y, . . . on the right (and only such terms) give rise to terms of order co in ^, 7?, . . . on the left. Hence, if A'l is the sum of all of the terms of order 0) of K, Ki(A, B,...;^,v,.. .)=A^Ki(a, b, . . . ; x, y, . . .), and A'l is a covariant. In this way, K = Ki-\-K2 + . . . . Henceforth, we shall restrict attention to covariants which are homogeneous in the variables, and hence of constant order. A covariant K of constant order oi of a single form f is homo- geneous in the coefficients, and hence of constant degree d. For, let / have the coefficients a, b, . . . and order p, and apply the transformation x = a^, y=aT], .... The coefficients of the resulting form are A =aPa, B=a^b Thus K(a^a,a^'b,. . . ; a-^x,a-^y,. . .)^(aiyK(a,b, . . . ;x,y, . . .)^ identically in a, a, b, . . . , x, y, . . . , since the left member 38 ALGEBRAIC INVARIANTS equals K(A, B, . . . ; ^, v, - - ■)- Now K is homogeneous in X, y, . . . , of order w; thus a-"K(aPa, aPb, . . .; X, y, . . .)=a^^K{a, b, . . . ; x, y, . . .). Thus if A' has a term of degree d in a, b, . . . , then a-"-aP'^=a«\ pd — (x} = q\, so that d is the same for all terms of A'. // / is a form of order p in q variables and if K is a covariant of degree d, order co and index X, then pd — w = q\. 22. Weight of a Covariant of a Binary Form. In f = aQxTP-\-paixP-^y-\-. . .-\-i.\aiX^-'y^+. . .+apyP the weight of at is k. We now attribute the weight 1 to :«; and the weight to y, so that every term of / is of total weight p. Apply to/ the transformation x=^, y=ar]. The hteral coefficients of the resulting form are ^0 = ^0, Ai=aai, . . ., Ap=aPap. If A is a covariant of degree d, order co, and index X, then K(Ao, . . . , Ap] ^, 7]) =a^K{ao, . . . , ap-, x, y). Any term on the left is of the form cAo'oAi'i . . . Ap'p^^yf {eo-\-ei+ . . . +ep=d). This equals coQ^mi^ . . . ap'vx'y'^-'a^-'' {W = r-\-ei+2e2+ . . . +pep). This must equal a term of the right member, so that W—<ji = \. But W is the total weight of that term. Hence every term of A is of the same total weight. A covariant of index X and order oj of a binary form is isobaric and its weight is co+X. For a form /of order p'm q variables, we attribute the weight 1 to Xi, Xi, . . . , ajj.i and the weight to Xq\ then (§ 17) every term of/ is of total weight p. By a proof similar to the above, a covariant of index X and order to of / is isobaric and its weight is w+X. §23] ANNIHILATORS OF CO VARIANTS 39 Consider a covariant K homogeneous and of total order w in the variables Xi, . . . , Xq of two or more forms /j. As in § 15, K need not be homo- geneous in the coefificients of each form separately, but is a sum of co variants homogeneous in the coefificients of each. Let such a. K he oi degree dt in the coefificients of /(, of order />j. As in ^21, ^pidt—u — qX. The total weight oi K is o} + \. For example, ii pi = po = q = 2, /i = aoX^+2aixy +a2y-, fi = boX^+2bixy+biy'^. The Jacobian of /i and fi is 4^", where K = {aohi — aibo)x^ + (aob2— a2bo}xy + (aib2 — a2bi)y''. Here di = d2=l, w = 2, X = 1 , and K is of weight 3. 23. Annihilators of Covariants K of a Binary Form. Pro- ceeding as in § 20, we have instead of (2) 9 AY /I A ■ t \ V ^KdAj dK d^.dKdv — K{Ao, . . . , Ap-y ^, v) = 2 — dn j=odAj dn d^ dn dv dn and obtain the follov^^ing result: K is covariant with respect to every transformation x=^ + «?7, >' = ??, if and only if it is annihilated by * (1) fi-3'l- ffi = ao-^+. . .^pa^-x^ 9^ \ 9a 1 9«p The binary form is unaltered if we interchange x and y, Gi and Qp-i ior i = 0, 1, . . . , p. Hence A' is covariant with respect to every transformation x=^, y = r}-\-fi^, if and only if it is annihilated by (2) 0-x^ (o^pa,^~Hp-l)a2^-+. • •+«.^). dy \ dao dai dcip-\/ Denote a covariant of order co of the binary p-'ic by K = Sx"+Six''-'^y-\-. . .+5.y'*'. * For another derivation, see the corollary in § 47. 40 ALGEBRAIC INVARIANTS By operating on K by (2), we must have {OS-Si)x''+{OSi-2S'2)x^-'y^. . . + (06',_] -co^Jx^-^ •1.1,- XT r-u +bS^r^O, identically in x, y. Hence A becomes (3) K = Sx"-hOSx'^-'y-\-hO^Sx''-Y + - • •+^0"^/', whik.by O6'. = 0, (4) O"+'6' = 0. Hence a covariant is uniquely determined by its leader S. (Cf. §25). Similarly, K is annihilated by (l) if and only if (5) 125 = 0, nSi = uS, 1252=(a;-l)5i, ..., QS^ = S^-i. The function 5 of ao, • • • , dp must be homogeneous and isobaric (§§21, 22). If such a function 6* is annihilated by Q, it is called a seminvariant. If we have S^, we may find 6*0,-1 by (5), then 8^-2, • • • , and finally Si. But if K is a covariant, we can derive S^ from S. For, by § 20, the transformation x=—-q, y=i replaces / by a form in which Ai = { — '[yap-u by the covariance of A', 6'U)r + - • .=6U)r + . . .^S{a)x'^-\-. . .+6'.(a)r, so that Soi{a) =S{A). Hence S^ is derived from 5 by the replacement (1) in § 20. When the seminvariant leader S is given, and hence also co (see Ex. 1), the function (3) is actually a covariant of/; likewise the function whose coefficients are given by (5). Proof will be made in § 25. In the following exercises, indirect verification of the covariance is indicated. EXERCISES 1. The weight of the leader 5 of a covariant of order w of a binary form / is W ~u = X and hence (§ 21) is ^(pd—w). Thus S and / determine u. 2. The binary cubic has the seminvariant S = aoa2 — ai''. A covariant with 5 as leader of is order to = 2 and is (aoflo— Ci^)^^ + (aofls — aia2)xy + (aiOs — a2-)y''. Since this is the Hessian of the cubic, it is a covariant. § 24] ALTERNANTS 41 3. Find the covariant of the binary cubic / whose leader is Co^fls— 3aoaitX2+2ai^ the only seminvariant of weight 3 and degree 3. It is the Jacobian of / and its Hessian. 4. A covariant of two or more binary forms is annihilated by 2fi-?/— , ZO-x—. dx dy 5. Find a seminvariant of weight 2 and partial degrees 1, 1 of a binary quadratic and cubic. Show that it is the leader of the covariant {aob2~2aibi +a-,bo)x + (aobi — 2ai&2 +a2bi)y. 24. Alternants. Consider the annihilators p p. p-i p) fi = 2 7(7^-1— =2 {k + l)at: 3=1 d(Jj k=o dOk + i = 2 {p-j-^i)aj-^^z' {p-k)a,+^:^ of invariants of a binary form. We have P f p, p-l ?^2 1 nO = ^ jaj-A(p-j + l)-^—^^ (p-k)a, + r-^\., P-I f p. P o2 00 = 2 (p-k)at + i\(k-\-\)^ — 1-2 jaj-i-^ — A.-=o [ 9at+i y=i 9d't9(/> The terms involving second derivatives are identical. Hence fiO-OS2 = 2 (i-\-l)(p-i)ai—-Zi{p-i-hl)ai^ i=o 9^1 »=i 9^ = 2 (p-2i)ai--, t=o 9«t since the first sum is the first sum in fiO with j replaced by i-\-l, and the second is the first sum in 00 with k replaced by i—1. If 6* is a homogeneous function of ao, . . . , Op of total degree d and hence a sum of terms cao'oai^i . . . Qp'v {eo-\-ei-\-. . .+ep = d), we readily verify Euler's theorem: «=o ddi 42 ALGEBRAIC INVARIANTS If 5 is isobaric, It is a sum of terms t = cao<^oai^i . . . dp^p (^1+2^2+. . .+pep=w) where w is constant; then P ■Ql P P g^ S idi — =2 iett = wt, S ?a, — = wS. % =0 9^1 % =0 »■ =0 9a< Hence if S is both homogeneous (of degree d) and isobaric {of weight w) in ao, . . . , dp, then (1) {W-0^)S = oiS, o} = pd-2w. A covariant with the leader S has the order w. (Ex. 1, § 23.) Since OS is of degree d and weight w-\-l, we have (1202 - 0212)5 = (120 - 012)05 +0(120 - 012)5 = (co - 2)05 +aj05 = 2(co- 1)05. Hence for r = 1 and r = 2, we have (2) (l20'--0^12)5 = r(co-r + l)0'-i5. To proceed by induction, note that (2) implies (120^+1 -0'-+il2)5= (120'--0'-12)05+0^(12C>-Ol2)5 = r(co-2-r+l)0'-5+coO'-5 = (r + l)(co-r)0''5, so that (2) holds also when r is replaced by r + 1. 25. Seminvariants as Leaders of Binary Covariants. Lemma. // 5 is a seminvariant, not identically zero, of degree d and weight w, of a binary p-ic, then dp — 2wl0. Suppose on the contrary that 5 is a seminvariant for which w<0, where u = dp — 2w. By the definition of a seminvariant, 125 = 0. Hence, by (2), § 24, (1) UO'S = r{o:-r + l)0'-''S (r = l, 2, 3, . . .) and no one of the coefficients on the right is zero. But being of degree d and weight dp + l; in fact, the largest weight of a function of ao, . . • , dp oi degree d is dp, the weight of a/. Then (1) for r = dp-wi-l gives O'*^-"'5 = 0. Then (1) §25] SEMIN VARIANT LEADERS OF COVARIANTS 43 for r = dp-w gives O''^""'-' 5 = 0, etc. Finally, we get 5 = 0, contrary to hypothesis. Theorem. There exists a covariant K of a binary p-ic whose leader is any given seminvariant S of the p-ic. The covariant K is in fact given by (3), § 23. By (1), for r = co4-l, fiO"+i5 = 0. Hence 0""'"^5' is a seminvariant of degree d and weight w' =w-\-ii}-\-\ =pd—w-{-l. Then dp-2w' = -{pd-2w)-2 is negative. Hence (4), §23, follows from the Lemma. Thus K is annihilated by the operator (2), § 23. Next, in ^-y^]K, the coefficient of x" y is r \w'S—-^U-r+\)0^-'S^-.W^S-r{u^-r+\)0:-'S\, which is zero by (1). Hence K is covariant with respect to all of the transformations Tn and T'n of § 19. Now r_irir_i = F: x=-y, y=x, as shown by eliminating ^, t], ^\, r]i between y= V, 1 j/ = »?i + ^i, 1 vi= y. Since K is of constant weight, it is covariant with respect to every St (§ 16). Hence, by § 19, K is covariant with respect to all binary linear transformations. 26. Number of Linearly Independent Seminvariants. Lemma. Given any homogeneous isobaric function S of ao, . . . , Gp of degree d and weight w, where co = dp — 2w>0, we can find a homogeneous isobaric function Si of degree d and weight w-\-l such that fi5i =5. 44 ALGEBRAIC INVARIANTS In (2), § 24, replace 5 by W'^S, whose degree is d and weight is w — r -}-l, so that its co is a)+2/' — 2. We get W^n'-'^S-0'-n'S^r{o:-\-r-l)0'-^2'-^S. Multiply this by (-1)'""^ ^ . ^ ^ r!co(a; + l) . . . {co-\-r-l) The new right member cancels the second term of the new left member after r is replaced by r — l in the latter. Hence if we sum from r = l to r = w-\-l, the terms not cancelling are those from the first terms of the left members, that from the right member for r = l, and that from the second term on the left for r=w-]-l. But the last is zero, since O'^+^6'^0, ff''S being of weight zero and hence a power of ao. Hence we get Q,Si=S, where r=irlco{cjo-\-l) . . . (co+r-l) Theorem.* The number of linearly independent seminvariants of degree dand weight w of the binary p-ic is zero if pd — 2w<0, but is {w; d, p)-(w-l; d, p), if pd — 2w't_0, where (w; d, p) denotes the number of partitions of w into d integers chosen from 0, 1, . . . , p, with repetitions allowed. If p^4, (4; 2, p) = S, since 4+0, 3 + 1, 2+2 are the partitions of 4 into 2 integers. Also, (3; 2, p) = 2, corresponding to 3+0, 2 + 1. Hence the theorem states that every seminvariant of degree 2 and weight 4 of the binary p-ic, p^i, is a numerical multiple of one such (see the Example in § 20). The literal part of any term of a seminvariant 6* specified in the theorem is a product of d factors chosen from ao, oi, . . . , Op, with repetitions allowed, such that the sum of the subscripts of the d factors is w. Hence there are {w; d, p) possible terms. Giving them arbitrary coefficients and oper- ating on the sum of the resulting terms with fi, we obtain a linear combination S' of the (w — 1; d, p) possible products * Stated by Cayley; proved much later by Sylvester. §27] LAW OF RECIPROCITY 45 of degree d and weight w-\. By the Lemma there exists* an 6" for which 9.S is any assigned S' . Thus the coefficients of our S'^9,S are arbitrary and hence are hnearly independent functions of the {w; d, p) coefficients of S. Hence the con- dition S25 = imposes (w-1; d, p) Hnearly independent linear relations between the coefficients of 5 and hence determines (w— 1; d, p) of the coefficients of S in terms of the remaining coefficients. Thus the difference gives the number of arbitrary constants in the general seminvariant S, and hence the number of linearly independent seminvariants S. 27. Hermite's Law of Reciprocity. Consider any partition W = «l + «2 + . . .+«5 of IV into b^d positive integers such that p'^ni^n2 ... ^ Wj. Write «i dots in a row; then in a second row write m dots under the first no dots of the first row; then in a third row write «3 dots under the first ns dots of the second row, etc., until w dots have been written in 8 rows. Now count the dots by columns instead of by rows. The number nn of dots in the first (left-hand) column is 8; the number m2 in the second column isj;«i; etc. The number of columns is wi ^ p. Hence we have a partition w = mi-\-m2-\-. . .-\-m^ of IV into TT ^ p positive integers not exceeding d. Hence to every one of the {w; d, p) partitions of the first kind corresponds a unique one of the {w; p, d) partitions of the second kind. The converse is true, since we may begin with an arrangement in columns and read off an arrangement by rows. The correspondence is thus one-to-one. Hence (w; d, p) = {w; p, d). By two apphcations of this result, we get (w; d, p)-{w-l; d, p) = (w; p,d)-iw-l; p,d). Hence, by the theorem of § 26, the number of linearly independent ♦Provided />(/-2(!ci-l)>0, which holds if /'(i-2i<:'^0. But if pd-2w<0, our theorem is true by the Lemma in § 25. 46 ALGEBRAIC INVARIANTS seminvariants of weight w and degree d of the binary p-ic equals the number of weight w and degree p of the binary d-ic. Let dp — 2w = o}^0. Then, by the theorem of §25, each seminvariant in question uniquely determines a covariant of order w. The number of linearly independent covariants of degree d and order w of the binary p-ic equals the number of linearly independent covariants of degree p and order w of the binary d-ic. The covariants are of course invariants if and only if = 0. EXERCISES 1. Show by means of (1), § 24, that w—hpd for an invariant. 2. Show that (6; 6, 3) = 7, (5; 6, 3) = 5. Find the two linearly inde- pendent seminvariants of weight 6 and degree 6 of the binary cubic. 3. There are only two linearly independent seminvariants of degree 4 and weight 4 of a binary quartic. Find them. 4. There is a single invariant or no invariant of degree 3 of the binary P-ic according as p is or is not a multiple of 4. (Cayley.) Hint: Every invariant of the binary cubic is a product of a constant by a power of its discriminant, of order 4 (§ 30). 5. The binary p-ic has a single covariant or no covariant of order p and degree 2 according as p is or is not a multiple of 4. (Cayley.) Hint: Every covariant of the binary quadratic /is of the type c D^f*, where c is a constant and D the discriminant of / (§ 29.) The degree 2«+w» of the product equals its order 2m if m = 2n. Thus / has a covariant of order and degree p if and only if ^ = 4«, viz., c Z)"/"". 6. No covariant of degree 2 has a leader of odd weight. 7. If 5 is of degree di in the coefficients of a binary pi-ic, of degree d2 in the coefficients of a p2-ic, . . . , and of total weight w, (2), §24, holds with fi and replaced by 212 and 20, and co replaced by I,pidi—2w. For any such S, there exists an ^i of partial degrees di and total weight w + 1 for which {ZQ)Si = S. If 5 is a seminvariant, co^O. Generalize §§ 26, 27, using (w; du pr, di, pu . . ) to denote the number of ways in which w can be expressed as a sum of di or fewer positive integers ^/>i, of di or fewer positive integers^/»2, etc. §28] FUNDAMENTAL SYSTEM OF COVARIANTS 47 Fundamental System of Covariants of a Binary Form, §§ 28-31 28. Certain Seminvariants. For ao9^0, we may set f = aoX^-]-paixP~'^y-{-. . .+apyP = ao(x-aiy) . . . (x-apy). Apply to / the transformation r„: x=^-\-nr], y = -q. Then each root aj of / = is diminished by w, since X — aiy= ^—(ai — n)r]. Hence the difference of any two roots is unaltered. In particular, if n=—ai/ao, f is transformed into the reduced form / . \ /-ao^" + ( w '2e-~n ^ + 1 IP] k'sf'-'TH where \ / \ f a' 2 = a'3- 03- -3""'^+2''C and the roots of/' = are ai+oi/ao (^' = 1, • • - ^ P) Since fli 'Za\ (ai— ai) + . . . + («!— ofp) aiH = a< — = — — — , ao p p each root of /' = is a linear function of the differences of the roots of /=0 and hence is unaltered by every transformation r„. The same is true of a'-i/aa, a'^/ao, . . . , which equal numerical multiples of the elementary symmetric functions of the roots of/' = 0. Hence the polynomials Ao^ ao'j'-z = aoao — ar, A.i^ao^a'3=ao^a3 — Saoaia2-{-2ai^, yl4 = ao'^<i'4 = flo'^«4 — 4ao^<iia3 +600^1^02 — 3a i^* are homogeneous and isobaric,* and are invariants of / with respect to all transformations r„. By definition they are, therefore, seminvariants of / provided the subscript of each A in question does not exceed p. * This is evident for A2, A3, At. Further A's will not be employed here. A general proof follows from § 34. 48 . ALGEBRAIC INVARIANTS Since /' was derived from / by a linear transformation of determinant unity, any semin variant S oif has the property S{ao, Hence any rational integral seminvariant is the quotient of a polynomial in ao, A2, . ■ . , ^j; by a power of gq. For /> ^ 4, we shall find which of these quotients equal rational integral functions of ao, . . . , Op and hence give rational integral seminvariants. The method is dua to Cayley. For p = l, S is evidently a numerical multiple of a power of ao. Since co is the leader of the covariant /=aox+aiy of /, we conclude that every covariant of a binary linear form / is a product of a power of / by a constant; in particular, there is no invariant. 29. Binary Quadratic Form. Since A2 does not have the factor ao, we conclude that every rational integral seminvariant is a polynomial in ao and A2. Now A 2 is an invariant of / (§4), and ao is the leader of the covariant/ of/. Hence a fundamental system of rational integral covariants of the binary quadratic form f is given hy f and its discriminant A2. We express in these words our result that any such covariant is a rational integral function of/ and A2. 30. Binary Cubic Form. We seek a polynomial P(ao, ^2,^3) with the implicit, but not expHcit, factor ao. Write A'l for the terms of Ai free of ao: (1) A'2 = -ai\ A'^ = 2a,K We desire that P(0, A' 2, ^'3) =0, identically in ai. Now 4^V+^V^0, (2) 4.A2^+Az^^ao''D, where D is the discriminant of the cubic form, D = a^rr.:r — Gaoaia2a3 +4aoa2^ +4ai^a3 — 3ai^a2^. §31j FUNDAMENTAL SYSTEM OF COVAKIANTS 49 By means of (2) we eliminate As~ and higher powers of A3 from P(ao, A2, A 3) and conclude that any semin variant is of the form w/ao'^, where tt is a polynomial in ao, A2, A3, D, of degree 1 or in yl 3. li k>0, we may assume that not every term of x has the explicit factor ao- In the latter case, t does not have the implicit factor ao- For, if it did, x' = 7r(0, A'o, A's, D')=0, D' = 4:ai^a3-Sai^a2^. Since as occurs in D', but not in A' 2 or ^'3, tt' is free of D'. By (1), the first power of A' 3 is not cancelled by a power of A' 2. Hence tt' is free of A' 3 and hence of A'2- A fundamental system of rational integral seminvariants of the binary cubic is given by ao, A2, A3, D. They are connected by the syzygy (2). A fundamental system of rational integral covariants of the binary cubic f is given by f, its discriminant D, its Hessian H, and the Jacobian J of f and H. They are connected by the syzygy (3) 4H^+J-^fW. The last theorem follows from the first one and (2), since ao, A2, A3 are the leaders of the covariants/, //, /. 31. Binary Quartic Form. We first seek polynomials P(ao, A2, A3, Ai) with the impHcit, but not explicit, factor oq. Thus r = P{0,A'2,A'3,AU)^0, A'2=-ai\ A'3 = 1ax\ A\= -Za^K The simplest P' is evidently 3^1 '2" +.4 '4. We get /44+3/l22 = ao-/, / = aoa4-4aia3+3a22. We drop Ai, and consider polynomials 7r(ao, Ao, A3, I) with the imphcit, but not exphcit, factor, ao. Such a polynomial is given by (2) , § 30. For ao = 0, D= -ai^I = A '2I. We have A2l-D = aQj, J = aoa2a4 — aoCa^ + 2aia2a3 — ai-a4 — ao^. Eliminating D between this relation and (2), § 30, we get (1) ao-V-ao2^2/ + 4^2^+^32 = 0. 50 ALGEBRAIC INVARIANTS In view of their origin, / and / are semin variants of the quartic /. Since they are unaltered by the replacement (1), §20, they are invariants of/ {cf. §20, Example and Ex. 1). In view of (I), tt equals a polynomial ^ in ao, A2, A^, I, /, of degree or 1 in ^3. Suppose that (/> does not have the explicit factor oq. Then the equal function of ao, . . . , a^ is not divisible by Qq. For, if it were, (/)(0, — fli^, 2ai3, 302^ — 4aia3, —ai^a^-]-. . .)— 0- In view of the term a^, <i> cannot involve /, and hence not /. Nor can </> be linear in ^3 in view of the odd power a\^. Hence is free of ^3 and hence of ^2. A fundamental system of rational integral seminvariants of the binary quartic is given by ao, A2, A3, I, J. They arc con- nected by the syzygy (1). A fundamental system of rational integral covariants of the binary quartic f is given by f, its invariants I and J, its Hessian H and the Jacobian G of f and H. They are connected by the syzygy (2) f^J-fmi-\-im-\-G^^O. The second theorem follows from the first one, since ao, A2, A3 are the leaders of the covariants/, H, G. It would be excessively laborious, if not futile, to apply the same method to the binary quintic, whose fundamental system is composed of 23 covariants,* most of which are very complex. The symbolic method is here superior both as to theory and as to compact notation (see Part III.). Canonical Form of Binary Quartic. Solution of Quartic Equations 32. Theorem. A binary quartic form f, whose discrim- inant is not zero, can be transformed linearly into the canonical form (1) X^ + Y^+QmX^YK * Faa di Bruno, Thcorie der Bindren Formen, German tr. by Walter, 1881, pp. 199, 316-355. Salmon, Modern Higher Algebra, Fourth Edition, 1885, p. 227, p. 347. §32] CANONICAL FORM OF QUARTIC 51 The reason there is here a parameter m lies in the existence of two invariants / and / of weights (and hence indices) 4 and 6, and hence a rational absolute invariant P/J-, i.e., one of index zero, and consequently having the same value for/ and any form derived from / by linear transformation. Since / vanishes for four values of x/y and hence is the product of four linear functions, it can be expressed (in three ways) as a product of two quadratic forms, say those in the right members of the next equations. To prove our theorem it suffices to show that there exist constant p, q, r, s (each 5^0) and a, /3 {a 9^0) such that p(x -]-ay)~-\-q(x-\-l3y)^ ^ ax^-\-2bxy-\-cy^, r(x +ay)- +5(x +/3y)- = gx^ + 2hxy + ky^. For, the product/ of these becomes (1) by the transformation A' = '^pr {x+ay) , Y = ^qs {x + ^y) , of determinant 5^0. The conditions for the two identities are p-\-q = a, pa-\-qfi = b, pa'^-\-q0^ = c, r-\-s = g, ra-\-s0 = k, ra^-{-s^^ = k. The first three equations are consistent if 1 1 a a ^ b ^{^-a)=C-b(a+l3)-\-aa^ = 0. If p = 0, or if q = 0, the same equations give b'~ = ac, so that the first quadratic factor of / and hence / would have a dcuble root. Similarly, the last three equations have solutions r^^O, ^-//(a+/3)+ga/3 = 0. If the determinant ah — bg is not zero, the last two relations determine a 4-/3 and a^, and hence give a and /3 as the roots of * {ah — bg)z- — (ak — cg)z-\-bk — ch = 0. * Its left member is obtained by setting x/y= — s in the Jacobian of the two quadratic factors of /. 52 ALGEBRAIC INVARIANTS If its roots were equal, the two relations would give C-2ba-\-aa^ = 0, k-2ha-\-ga^ = 0, and the two quadratic factors of/ would vanish for x/y= -a. If ah-bg = 0, but ch — bk^^O, we interchange x with y and proceed as before. If both determinants vanish, either b9^0 and the second quadratic factor is the product of the first by h/b, or else b = and hence h = and no transfor- mation of/ is needed. 33. Actual Determination of the Canonical Quartic. Let A denote the determinant of the coefficients of x, y in A', Y. Then /, its invariants / and / and Hessian H are related to the canonical form, its invariants and Hessian, as follows: / = A4(l+3m2), J^A^(m-m^), Zr = A2iw(X4+F^) + (l-3w2)X2 72|_ Thus A^m may be found from the resolvent cubic equation 4(A2w>^-/(A2w)+/ = o. Then A* may be found from /. We may select either square root as A2 and hence fmd m. In fact, by replacing .Y by XV — 1 in /, the signs of A2 and m are changed. By elim- inating X^+F^, we get If 9^2=1, / is the square of X'^dzY'^ and the discriminant of / would vanish. Hence we obtain XY by a root extraction. Thus X and F are determined up to constant factors / and t'^. We may find / by comparing the coefficients of .t^ and x^y in / and the expansion of its canonical form, or by use of the Jacobian G oi f and H: G = ^K\-9m^~)XY{X^-Y^), and combining the resulting X*— F^ with the earlier A^^-f-F'*. Or from/ and XY we can find A'2 + F2 and then X±F. To solve / = 0, we have only to find the canonical form §34] SEMINVARIANTS IN TERMS OF THE ROOTS 53 Seminvariants, Invariants, and Covariants of a Binary Form/ as Functions of the Roots of/=0, §§ 34-37. 34. Seminvariants in Terms of the Roots. Give / the nota- tion used in § 28, so that ai, . . . , ap are the roots of /=0. After removing possible factors ^o from a given seminvariant of /, we obtain a seminvariant 5 not divisible by qq. Let 5 be the degree of the homogeneous function S of the a's. Thus S is the product of a^ by a polynomial in ai/ciQ, . . . , dp/ao of degree 8. The latter equal numerical multiples of the ele- mentary S}'mmetric functions of ai, . . . , ap, each of which is linear in every root. Hence our polynomial equals a sym- metric polynomial a in ai, . . . , ap of degree 5 in every root. Since S is of constant weight w and since at/ao equals a function of total degree i in the roots, a is homogeneous in the roots and of total degree w in them. Besides being homogeneous and isobaric in the a's, a sem- invariant must be unaltered by every transformation Tn of § 28. Under that transformation, each root is diminished, by w (§28). Since ai = ai-{-{ai — ai) (^ = 2,. . . , p) we can express o- as a polynomial P(ai) whose coefficients are rational integral functions of the differences of the roots. If P{ai) is of degree ^1 in ai, we have P(ai)=P{ai — tt), for all values of ?i. But an equation in n cannot have an infinitude of roots. Hence P(ai) does not involve ai, so that a equals a polynomial in the differences of the roots. Multiplying by the factors ao removed, we obtain the theorem : Any seminvariant of degree d and weight w of the binary form aQX^-\-. . . equals the product of a(f by a rational integral symmetric function a of the roots, homogeneous {of total degree w) in tJie roots, of degree ^ d in any one root, and expressible as a polynomial in the differences of the roots. Conversely, any such product can be expressed as a poly- nomial in the a's and this polynomial is a seminvariant. 54 ALGEBRAIC INVARIANTS Since the factor o- is symmetric in the roots, and is of degree ^d in any one root, its product by ao'^ equals a homogeneous polynomial in the a's whose degree is d. This polynomial is isobaric since a is homogeneous, and is unaltered by every transformation T„, since o- is expressible as a function of the differences of the roots. The importance of these theorems is due mainly to the fact that they enable us to tell by inspection (without com- putation by annihilators) whether or not a given function of the roots and ao is a seminvariant. A like remark applies to the theorem in § 35 on invariants and that in § 36 on covariants. EXAMPLE The binary cubic has the seminvariant ao*2(ai— a2)(ai — a3)=aoK^«i"~2:aia2) 3 = CoM(2ai)='-3Saia2}=ao-| ( — -j -sl — ) i = -9{aoa2-ai^). 35. Invariants in Terms of the Roots. A seminvariant of / is an invariant of / if and only if it is unaltered by the trans- formation X = - 77, >» = H§ 20) . For the latter, X — ay= —al ^H — r?j, so that ar is replaced by — I far, and hence ar-as by ar — as (XrOCs The coefficient of ^^ in the transformed binary form is Ao = ( — iyaia2 . . . ocpao. By § 34, any seminvariant of / is of the type ao'^2ci(product of w factors like ar — as). Hence this is an invariant if and only if it equals (^-iyd(^ai . . . a:p)'^ao'^Sct( product of the w corresponding-^^ §35] INVARIANTS IN TERMS OF THE ROOTS 55 and hence if ±0:1'' . . . ap'^ equals the product of the factors arOis in the denominators. This is the case if and only if each root occurs exactly d times in every term of the sum and if pd is even. By the total number of a's, pd = 2w. Any invariant of degree d and weight w of the binary form aoX^-\- . . . equals the product of ai,^ by a sum of products of constants and certain dij'erences of the roots, such that each root occurs exactly d times in every product; moreover, tJie sum equals a homogeneous symmetric function of the roots of total degree w. Conversely, the product of any such sum by ao'^ equals a rational integral invariant. EXERCISES 1. a(,^{ai—a2)- is an invariant of the binary quadratic form. Any invariant is a numerical multiple of a power of this one. 2. ai,-2(ai— a2)'(«3— 04)' is an invariant of the binary quartic. 3. 00^2(01— a2)(ai— a:)) is not an invariant of the binary cubic. 3 4. If we multiplyrtn^'''"^' by the product of the squares of the differences of the roots of the binary p-ic f, we obtain an invariant (discriminant of /) . Also verify that pd = 2w. 5. The sum of the coefficients of any seminvariant is zero. Hint: Use/= (x + y)^, whose roots are all equal. 6. Every invariant of the binary cubic is a power of its discriminant. 7. A function which satisfies the conditions in the theorem of § 35 except that of symmetry in the roots is called an irrational invariant. If aj, . . . , a4 arc the roots of a binary quartic/, and U=(ai—ai)(a-i—a3), V= {a> — ai)(a:i—ai) , W= {ai—a2)(a3—a4) , why are aou, a„v, aoW irrational invariants of /? They are the roots of z' — 12/2— 5 = 0, where 5- is the product of ao" by the product of the squares of the differences of the roots and hence is the discriminant of/. Hints: tt+Z)+w = 0, and s = nv-\-iiw-\-mv is a symmetric function of ai, . . . , at in which each aj occurs twice in every product of differences, so that ao'^s is an invariant of degree 2. By the Example in §20, ao^s = cI, where c is a constant. To determine c, take ai = l,a2= — 1, 03 = 2, «<= — 2, so that /=(x2-y2)(;t2-4y2), 7 = 73/12, u=-Q, v=l, w = 8, 5= -73. Hence c= — 12. As here, so always an irrational algebraic invariant is a root of an equation whose coefficients are rational invariants. 56 ALGEBRAIC INVARIANTS 8. If ai, oil are the roots of the binary quadratic form/, and as, a^ the roots of /' in § 11, the simultaneous invariant ac' +a'c — 2hh' = aa'lasat +aia2 — K«i +"2) (as -\-o!i) \ = \ao{u — v) , if the product jf is identified with the quartic in Ex. 7. Hence a simul- taneous invariant of the quadratic factors of a quartic is an irrational invar- iant of the quartic. Why a priori is the invariant three-valued? 9. The cross-ratios of the four roots of the quartic are —v/ii, etc. These six are equal in sets of three if 7 = 0. For, if 5 = 0, , , . , —V —u —w vw — u\—v—w) = u^, uw = v{-'u—w) = v^, — = — — . U W V The remaining three are the reciprocals of these and are equal. 10. By Ex. 3, § 11, one of the cross-ratios is —1 if ac'-f- . , . =0. Why does this agree with Ex. 8? 11. The product of the squares of the differences of the roots of the cubic equation in Ex. 7 is known * to be — 4.(-12l)^-278'- = aoKti-v)-{n-w)-(v-wy. Also,* 52 = 256(7' -27/2). Hence the left member becomes 3^-4^/2. Thus 33. 42/= zizaQ^{u—v)(u—w){v—w). Using J from § 31, and the special values in Ex. 7, show that the sign is plus. Verify that the cross-ratios equal —1, —1, 2, 2, 5, 5, if 7 = 0. 36. Covariants in Terms of the Roots. Let K (ao, ■ . . ,ap\x, y) be a covariant of constant degree d (in the coefficients) and constant order co (in the variables) of the binary form/ = aoa;^ + . . . Then K = Qo'^fK, where k is a polynomial in x/y and the roots a\, . . . , ap oi /=0. Under the transformation Tn in §28, let / become ^0^^+ . . . , with the roots a'l, . . . , a'p. Then X ? / II a% = cc J, aT — (Xs=OLT — OLs. y V Making use of the identities y \y Cf. Dickson, Elementary Theory of Equations, p. 33, p. 42, Ex. 7. §36] COVARIANTS IN TERMS OF THE ROOTS 57 we see that k equals a polynomial P(ai) whose coefficients are rational integral functions of the differences of x/y, ai, . . . , ap in pairs. Since K{Ao, . . . , Ap] ^, ri)=K{ao, . . . ,ap; X, y), Ao = ao, r] = y, we have ^1 a'l, . . . , a'p, -j =«( ai, . . . , ap, -j. The left member equals -P(a'i) since a'i = {ai — ai)-i-a'i, -=( ai)-\-ai. V \y / Hence P(ai-n)-P(ai)=0 for every n. Hence ai does not occur in P(ai), and k is a polynomial in the differences of x/y, ai, . . . , ap. Let W be the weight of K and hence of the coefficient of y. Then k is of total degree W in the a's and of degree co in x/y. Thus K = 2C(| product of w differences hke — aA [ y J •{product of W — o) differences like ar—as]. Hence K = ao'^ZCi\'prod\iQt of w differences like x—ary] • {product of W—oi differences like ar—as\. Next, for x=—t], y=^, f becomes F = Ao^^+ . . . with a root — X/ut corresponding to each root ar of /. The function K for F is product of CO differences like ^-\ — ri = - !^[ ar —ar \ • ] product of W — o) differences like — ^ [ . \ aras J Using the value of Aq in § 35, we see that the factor \-\yW . . . ap'^ must be cancelled by the —ar and the aras in the denominators. 5S ALGEBEAIC INVARIANTS Thus each term of the sum involves every root exactly d times. The signs agree since as follows by counting the total number of as. Any covariant of degree d, order co and weight W of ao{x—aiy) . . . (x—apy) equals the product of ao^ by a sum of products of constants and w differences like x—ary and W — oj differences like ar—as, such that every root occurs in exactly d factors of each product; more- over, the sum equals a symmetric function of the roots. Conversely, the product of ao^ by any such sum equals a rational integral covariant. EXERCISES 1. j=aoX^-\-ZaiX'^y-\-Za2xy-+a3y^ has the covariant K = ao'^'L{x-aiyy{oL'z-azy-. 3 Show that the coefficient of x"^ in K equals — 18(aoa2— Oi^). Why may we conclude that K= — 18H, where H is the Hessian of/? 2. The same binary cubic has the covariant ^ ao^Z(x—aiy){x—a2y){a2 — a3)ia3—ai) = 9H. 3 3. Every rational integral covariant of the binary quadratic / is a prod- uct of powers of / and its discriminant by a constant. 37. Covariant with a Given Leader S. If the seminvariant S has the factor ao, and S = aoQ, and if Q is the leader of a covariant K of /, then, since ao is the leader of /, S is the leader of the covariant fK. Hence it remains to consider only a seminvariant 6* not divisible by oq. li S is of degree d and weight w, S = ao'^XCiiproduct of w factors like ar—as), where each product is of degree at most d in each root, and of degree exactly d in at least one root (§ 34). If each product is of degree d in every root, S is an invariant (§ 35) and hence is the required covariant. In the contrary case, let a2, for example, enter to a degree less than d; we supply enough factors x—a2y to bring the degree in a2 up to d. Then ao'^ §38] DIFFERENTIAL OPERATORS 59 multiplied by the sum of the total products is a covariant with the leader S. For example, ao^2(a2— 03)^, 00^2(0:2 — 0:3) («3—ai) 3 3 are the leaders of the covariants in Exs. 1, 2, § 36, of the binary cubic. The present result should be compared with the theorem in § 25. We may now give a new proof of the lemma in § 25 that dp — 2w^0 for any semin variant 5 of degree d and weight ■w of the binary p-ic. Whether 5 has the factor ao or not, the first term of the resulting covariant K is Sx", where oo = dp — 2w. For, in each product in the above S, the roots ai, . . . , ap occur 2w times in all. In K each root occurs d times. Hence we inserted dp — 2w factors x—ay in deriving K from S. 38. Differential Operators Producing Covariants. Let the transformation T: x = a^-\-^v, y = y^-\-8v, A=a8-^y^0 replace /(x, )') by (/)(^ r?). Then d±^^dx dldy^^df^ df d^ dx d^ dy d^ dx dy d±^^dx_^^dy^^d[_^^^ di dx dv dy dv dx dy Solving, we get dy dv d^ dx dv d^ or df=D<p, dif = D]_(p, if we introduce the differential operators dy dx dv 9? a^ a^ As usual, write d'-dif for d\d{dif)]. Since the result of operating with d on df is the same as operating with D on the equal function D^ of ^ and r], we have d-f=D"4>. Similarly, 60 ALGEBRAIC INVARIANTS (r+5 = a)), The right member is the result of operating on with the operator obtained by substituting D for d/dn and Di for — d/dk in whose terms are partial derivatives of order w. Hence, if the form becomes X(^, r?) under the transformation T, our right mem- ber is the result of operating on ^ with \{d/drj, — d/9^). The left member is the result of operating on / with \ dy dx) • \dy dxj Hence if T replaces the forms J {x, y), l{x, y) by 0(^, rf), X(^, rj), then L VaV 9^/ 0(?, ^)=A'' f{x, y) \-\dy dxj \ is a consequence of the equations for T, if oj is the order of l{x, y). Let / and / be covariants of indices m and n of one or more binary forms ft with the coefficients ci, C2, . . . . Under T let the transformed forms have the coefficients Ci, C2, .... Then /(C; ^, v) = A-/(c; x, y), 1{C; ^, v) = A«/(c; x, y). But 0(^, ri) =f{c\ X, y), by the earlier notation. Hence ^{^, r})=A-^f{C; ^, v), X(^, 7;)=A-"/(C; ^, r?). Inserting these into the formula of the theorem, and mul- tiplying by A"'+'', we get ['{''■• i'-i-^\'''-'''^^^''^^'ii'-4y'-l^) f{c;x,y). The function in the right member is therefore a covariant of index w+w+w of the ft. We therefore have the theorem of Boole, one of the first known general theorems on covariants: §38] DIFFERENTIAL OPERATORS 61 Theorem. If I and J are any covariants of a system of binary forms, we obtain a covariant {or invariant) of the system of forms by operating on f with the operator obtained from I by replacing x by d/dy and yby— d/dx, i.e., x'"/ byf — iyd'"^^/dy''dx\ EXERCISES 1. Taking l=f=ax-+2bxy+cy-, obtain the invariant 4{ac—b-) of /. 2. If /=/ is the binary quartic, the invariant is 2 -4! / of § 31. 3. Using the binary quartic and its Hessian, obtain the invariant /. 4. Taking l = aoX^+. . . , f=boX^+. . ., obtain their simultaneous invariant If also /=/, we have an invariant of/, which vanishes if p is odd. For /> = 2 and ^ = 4, deduce the results in Exs. 1 , 2. 5. A fundamental system of covariants of a quadratic and cubic Q = Ax^ +2Bxy+Cy, f= ax' +3bx-'y +3cxy^- +dy^ is composed of 15 forms. We may take Q and its discriminant AC—B''; f, its discriminant and Hessian h, given by (5) and (2) of § 8, the Jacobian /of/and£?: J=(d''d-3abc+2b')x^+3(abd+b'-c-2ac^)x'-y +3(2b''d-acd-bc")xy'' + {3bcd-ad'-2c')y»; the Jacobian of /and Q: {Ab-Ba)x' + i2Ac-Bb-Ca)x^y + (Ad+Bc-2Cb)xy^ + (Bd-Cc)y^; the Jacobian of Q and //: {As-Br)x^ + {At-Cr)xy + (Bt-Cs)y'-; the result of operating on / with the operator obtained as in the theorem from l=Q: Li = (aC+cA-2bB)x + (bC+dA-2cB)y; the result of operating on Q with the operator obtained from Lr. L2= {aBC-b{2B^+AC) +3cAB-dA']x + \aO-3bBC+c(AC+2B^~)-dAB\y; 62 ALGEBRAIC INVARIANTS the result Lz, of operating on / with Q and the result Li of operating on Q with Li (so that Li and Li may be derived from Ly and L2 by replacing a, . . . , d by the corresponding coefficients of J) ; the intermediate invariant At+Cr—2Bs of Q and h (§ 11); the resultant of Q and/: c2C3-6a65C2+6acC(252-.4C)+a(f(6.45C-853) +96^.4 C2 -18bcABC+6bdA(2B^-AC)+9cW-C-()cdBA^+d\L^; the resultant of Zi and Z4 ( = resultant of Z,2 and L3), obtained at once as a determinant of order 2. Salmon, Modern Higher Algebra, § 198, gives geo- metrical interpretations. Hammond, Arner. Jour. Math., vol. 8, obtains the syzygies between the 15 covariants. PART III SYMBOLIC NOTATION The Notation and its Immediate Consequences, §§ 39-41 39. Introduction. The conditions that the binary cubic (1) f^aoXi^-\-3aiXi"X2-{-3a2XiX2^-\-(i3X2^ shall be a perfect cube (2) (aiXi -\-a2X2y are found by eliminating ai and a2 between (3) ai'^ — Qo, ai-a2 = ai, axa2~ — CL2, a<^=az, and hence the conditions are (4) aoa2 = ai~, aia3=a2^. Thus only a very special form (1) is a perfect cube. However, in a symbolic sense * any form (1) can be rep- resented as a cube (2), in which ai and 0:2 are now mere symbols such that (3') ai'^, ci^a.2, aiQ:2~, 0^2 are given the interpretations (3), while any linear combination of these products, as 2ai^ — la2^, is interpreted to be the cor- responding combination of the a's, as 2aQ — la2,. But no inter- pretation is given to a polynomial in ai, 012, any one of whose terms is a product of more than three factors a, or fewer than three factors a. Thus the first relation (4) does not now follow from (3), since the expression 0:1^0:2" (formerly equal to both * Due to Aronhold and Clebsch, but equivalent to the more complicated hyperdeterminants of Cajdey. 63 64 ALGEBRAIC INVARIANTS Gofl2 and Gi^) is now excluded from consideration; likewise for ai^ao^ and the second relation (4). In brief, the general binary cubic (1) may be represented in the symbolic form (2) since the products (3') of the symbols ai, 02 are in effect independent quantities, in so far as we permit the use only of linear combinations of these products. But we shall of course have need of other than linear functions of oo, • . .,03- To be able to express them sym- bolically, we represent / not merely by (2), but also in the symbolic forms (5) (/3iXi+/32^2)^, (71^1+72:^2)^, . . ., so that (6) /3i3 = ao, |8i2/32 = ai, Pip2^ = a2, /32^ = a3; 71^ = ^0, .... Thus aoCf2 is represented by either ai^/3i/32- or jSi^q;iq:2^, while neither of them is identical with the representation ara2^i^^2 of ai^. Hence aoa2-ai2 = i(ai3^i/322+|3i3aia22-2ai2a2/3i2^2) = iai/3i(ai^2-a2/3i)2. We shall verify that this expression is a seminvariant of /. If Xi^Xi-{-tX2. .r2=A^2, then /becomes F = AoXi^-{-. . . , where Ao = ao, Ai=ai-\-tao, A2 = a2-\-2tai-\-fiao, ^3 = a3+3/a2+3/2ai+/3ao. Hence, by (3), F = {aiXi-{-a 2X2)^, a 2=oc2-\-tai. Similarly, the transform of (5i) is (iSiXi +^'2X2)3, |8'2 = i82+^i8i. Hence we obtain the desired result ^0^2-^i2 = |ai/3i(ai^'2-«'2/3i)2 §40] SYMBOLIC NOTATION 65 40. General Notations. The binary n-ic is represented symbolically as a^" = /3x" = . . . , where ax = aiXi-\-a2X2, /3j; = /3i.ri +182X2, . • • , a2" = a„; /3i" = ao, .... A product involving fewer than n or more than n factors ai, 0:2 is not employed except, of course, as a component of a product of n such factors. The general binary linear transformation is denoted by T: xi = ^iXi + -niX2, X2 = aA^i + 7,2X2, {kri)9^0 1 where {^n) = hm— hvi- It is an important principle of com- putation, verified for a special case at the end of § 39, that T transforms ax" into the »th power of the linear function (ai ^1 -\-0c2 h)X\ + (ai 7/1 +a:2r?2)-X'2 =a^Xi +a,X2, which is the transform of a^ by T. Further, /j\ «£ oir, ^ ai a2 h V\ /?€ ^n iSi 182 * ^2 7/2 where (ajS) =ai/32 — q:2/3i = — (/3a). Thus (a^^,-a,^J)" = (^7,)"(a|8)^ so that (ai3)" is an invariant of ai"=fij^ of index «. Since {^aY represents the same invariant, the invariant is identically zero if n is odd. = M)(l^), EXERCISES 1. {a^y is the invariant 2(aoa2 — ar) of ccx- = 0x'^- 2. (a/3)^ is the invariant 2/ of ax' = /Ji* (§ 31). 3. (ai3)2 ((37)2 (Ya)2 is the invariant 6/ of ax* = /3i<=7i^ (§ 31). 4. The Jacobian of ax"* and /3x" is 7" — ! ";3x""Vi 7W — 1 «|3x" ~^/32 :m«(«/3)ax'"-Vx'^-^ 66 ALGEBRAIC INVARIANTS -5. The quotient of the Hessian of ax" = 0x" by «'(« — 1)^ equals n —2 9 ax Oil |3/ "-/i,/32 a 102 &x n-2 010: one-half of the sum of which equals § a^" ^/3j;" '^{a0)'^. 6. ai Pi 7i "2 ft 72 az ft yx = M 7:t + (^7)«:r + (7«)ft = 0. 41. Evident Covariants. We obtain a covariant iiC of / = «/^ = /3/ = . . . by taking a product of w factors of type ax and X factors of type (a(S), such that a occurs in exactly n factors, jS in exactly n factors, etc. On the one hand, the product can be inter- preted as a polynomial in ao, • • • , a», ^i, ^2- On the other hand, the product is a covariant of index X of /, since, by (1), §40, {ABy{Acy{Bcy . . . a^b'^c^ . . . = i^rjy{a^y{ayy{0yy . . . ax«i8.^7/ • • • , if \ = r-\-s-\-t-\-. . . and Ax = AiXi-{-A2X2, Ai=a^, ^2=«„ {AB)=AiB2-A2Bi, etc. The total degree of the right member in the as, /3's, . . . is 2\-\-o: = nd, if d is the number of distinct pairs of symbols Oil, a2; /3i, /32; . • .in the product. Evidently d is the degree of A' in ao, ai, • • • , and co is its order in xi, X2. Any linear combination of such products with the same CO and X, and hence same d, is a covariant of order co, index X and degree d of /. EXERCISES 1. {a0){ay)ax^l3x'yx*and {ai3y{ay)ax'tix'y'x are CO variants ofax^=ft^= 7x^ 2. {al5)W "'■ fix"" "'■ is a covariant of ax'\ ft^. 3. Iim = n, ft"=aa;"^ and ris odd, the last covariant is identically zero, 4. aoXi^+2aiXiX-+aoXo^ and boXi^+2biXiX2+b2X2^ have the invariant §42] SYMBOLIC NOTATION 67 COVARIANTS AS FUNCTIONS OF TwO SYMBOLIC TypES, §§ 42-45 42. Any Co variant is a Polynomial in the ax, (a^). This fundamental theorem, due to Clebsch, justifies the symbolic notation. It shows that any covariant can be expressed in a simple notation which reveals at sight the covariant property. While a similar result was accomplished by expressing covariants in terms of the roots (§36), manipulations with symmetric functions of the roots are usually far more complex than those with our symbolic expressions. The nature of the proof will be clearer if first made for a special case. The binary quadratic ax^ has the invariant K = aoa2—ai~ of index 2. Under transformation T of § 40, ai^ becomes (a{Xi+a,A'2)^=/loA'r + . . . , Ao=a(^, Ai=a^a^, A2=a^^. Hence AoAo—Ar equals We operate on each member twice with (1) v = -^- ^ 9^i9'72 9^29771' and prove that we get Q{a0)^ = l2K, so that K is expressed in the desired symbolic form. We have (^^) = ^l'72-^2 VI, :^(^r?)2 = 2(^77)a, -r^(^vy = 2i^v)+2n^^, dV2 0K10V2 OVl dK20r]l V{^vy = Q{^v), F2(|r,)2 = 12, since F(^r?) =2, by inspection. Next (2) Va^^r, = V{ai ^1 +a2 ^2) (Pi Vi +132^72) =«il82 -a2^i = (oc0). 68 ALGEBRAIC INVARIANTS Hence = /3ea,(a^)+aj^,(i3«), The difference of the expressions involving V^ is 6(q:/3)2. Hence if (1) operates twice on the equation preceding it, the result is 43. Lemma. F'^(^t7)" = (w + 1)(w!)2. We have proved this for w = 1 and n = 2. li n'L2, 9?i9'72 Similarly, or by interchanging subscripts 1 and 2, we get Subtracting, we get F(^T?)" = !2w+w(w-l)}(^r7)"-i=w(;^ + l)(^r7)"-i. It follows by induction that, if r is a positive integer, F'-(^7?)"=(w + l)lw(fi-l) . . . (w-r+2)|2(w-r+l)(^7J)"-^ The case r = n yields the Lemma. 44. Lemma. // the operator V is applied r times to a product of k factors of the type aj and I factors of the type /3„ there results a sum of terms each containing k—r factors a^, l—r factors )3„ and r factors {a0) . The Lemma is a generalization of (2), § 42. To prove it, set §44] SYMBOLIC NOTATION 69 Then (s) a (<)' a?29^i . = 1^ = 1 aj(^^/3,^'^' Subtracting, we get Hence the lemma is true when r = l. It now follows at once by induction that (1) VAB A B = S2(a(si)|3('i)) . . . {a(sr)^{tr)). «l (si) . . . ajCsO |3,('i) . . . |S,('0' where the first summation extends over all of the hik — l) . . . (^' — r + 1) permutations ^i, . . . , ^r of 1, . . . , ^ taken r at a time, and the second summation extends over all of the /(/ — I) . . . (/ — r + 1) permutations /i, . . . , Ir of 1, . . . , / taken ;' at a time. Corollary. The terms of (1) coincide in sets of rl and the number of formally distinct terms is II 1 /k\/l\ , rl. {k-r)\ {l-r)\ rl \r/\r For, we obtain the same product of determinantal factors if we rearrange si, . . . , Sr and make the same rearrangement of /i, . . . , tr. 45. Proof of the Fundamental Theorem in § 42. Let K be a homogeneous covariant of order cu and index X of the binary form / in § 40. By § 40, the general linear transformation replaces /=a:x" by (1) ^t = ai"-V (^ = 0,1,. . .,n). k=o\k Hence 70 ALGEBRAIC INVARIANTS By the covariance of K, (2) K{Aq, . . . ,An\ Xi, X2) = {h)^K{aQ, . . . , a„; :ki,:^2). By (1) the left member equals »'=0 in which the inner summation extends over various products AB, where ^ is a product of a constant and factors of type a|, and 5 is a product of a constant and factors of type a,. Let Xi=y2, and X2^—yi. Then, by solving the equations of T, § 40, Xi=yJ{^v), X2=-yi/{^ri). Hence the equation (2) becomes i 2{-iyABy„'^-'yii={^vy+''K. t=0 Since the right member is of degree X + co in ^i, ^2, and of degree \-\-co in 771, 772, we infer that each term of the left mem- ber involves exactly X + co factors with subscript ^ and X+w factors with subscript rj. Operate with F^"*"" on each member. By § 43, the right member becomes cK, where c is a numerical constant 5^0. By § 44, the left member becomes a sum of products each of X+co determinantal factors of which co are of t^'pe {ay) = ax, and hence X of type {a0). The last is true also by the definiton of the index X of K. Hence K equals a polynomial in the symbols of the types ax, (a0). To extend the proof to covariants of several binary forms a/, yx^, . . . , we employ, in addition to (l),Ct = 7j*"~*7,*, • • • and read a^, y^, . . . for a^ in the above proof. FiNITENESS OF A FUNDAMENTAL SYSTEM OF COVARIANTS, §§ 46-51 46. Remarks on the Problem. It was shown in §§28-31 that a binary form / of order <5 has a finite fundamental system of rational integral covariants A'l, . . . , Ks, such therefore that any rational integral covariant of / is a poly- § 47] FINITENESS OF COVARIANTS ; 71 nomial in Ki, . . . , Kg with numerical coefl&cients. We shall now prove a like theorem for the covariants of any system of binary forms of any orders. The first proof was that by Gordan; it was based upon the symbolic notation and gave the means of actually constructing a fundamental system. Cayley had earlier come to the conclusion that the fundamental system for a binary quintic is infinite, after making a false assumption on the independence of the syzygies between the covariants. The proof reproduced here is one of those by Hilbert; it is merely an existence proof, giving no clue as to the actual covariants in a fundamental system. 47. Reduction of the Problem on Covariants to one on In- variants. We shall prove that the set of all covariants of the binary forms /i, ...,/* is identical with the set of forms derived from the invariants 7 of /i, . . . , ft and l^xy' — x'y by replacing .r' by x and y' by y in each /. It is here assumed (§ 15) that 7 is homogeneous in the coefficients of / and that the covariants are homogeneous in the variables. Let the coefiicients of the /'s he a, b, ... , arranged in any sequence. Let A, B, . . .be the corresponding coefficients of the forms obtained by applying the transformation in § 5. The latter replaces / by W-^'t], where r,'=ay'-yx', ^' = bx'-^y'. Solving these, we get Ax'=a^'+^r?', ^y' = yt'J^bn'. Let I{a, b, . . . ; x', y') be an invariant of / and the /'s. Then I(A,B,. . .; ^\r,')=AH(a,b,. . .; x',y'). Since 7 is homogeneous, of order w, in x', y', the right member equals A^-"7((Z, b, . . . ; Ax', Ay')- Hence we have the identity in ^', rj' : liA, B,. . .; ^', r?0^A^-<::7(a, b, . . . ■ a^'+/3r?', y^'+W). 72 ALGEBRAIC INVARIANTS Thus we may remove the accents on ^', t] . Then, by our transformation, I{A,B,. . . ; ^,v)=A^-"I{a,b,. . .; x,y). Hence I {a, b, . . . ; x, y) is a. covariant of /i, . . . , ft oi order o) and index X — co. * The argument can be reversed. Note that the sum of the order and the index of a covariant is its weight (§ 22) and hence is not negative. Corollary. A covariant of the binary form / has the annhilators in § 23. For, an invariant of / and xy' — x'y has the annihilators fi-y--, o-x'^, ax" ay' 48. Hilbert's Theorem. Any set S of forms in xi, . . . , Xa contains a finite number of forms Fi, . . . , F^ such that any form F of the set can be expressed as F=fiFi-\-. . .4-/t^t, where fx, . . . , fk drc forms in xi, . . . , Xn, but not necessarily in the set S. For w = l, 5 is composed of certain forms Cix^\ C2X^\ .... Let Cs be the least of the e's, and set Fi=CsX^K Then each form in 5 is the product of Fi by a factor of the form cx«, e ^ 0. Thus the theorem holds when n = l. To proceed by induction, let the theorem hold for every set of forms in n — \ variables. To prove it for the system S, we may assume, without real loss of generality,* that S contains a form Fq of total order r in which the coefficient of xj is not zero. Let F be any form of the set 5. By division we have F = FqP-\-R, where i? is a form whose order in Xn * Let F be a form in 5 not identically zero and let the linear transformation Xi = Cii3'i+Cj2>'2+. . .+Cinyn (/=L. . . , «) replace F(:x:i, . . . ,Xn) by K{yi, . . . , jn)- In the latter the coefficient of the term involving only >'« is obtained from F by setting Xi = cin and hence is F{cin., con, ■ ■ • , Cnn), which is not zero for suitably chosen c's (Weber's Algebra, vol. T, p. 457; second edition, p. 147). But our theorem will be true for 5 if proved true for the set of forms K. §48] FINITENESS OF COVARIANTS 73 is <r. In i? we segregate the terms whose order in x» is exactly r — 1, and have where M is a form in xi, . . . , a-„_i, while iV is a form in iCi, . . . , Xn whose order in Xn is :^ /- — 2. Each F uniquely deter- mines an M. For the defmite set of forms M in w — 1 variables the theorem is true by hypothesis. Hence there exists a fmite number of the M's, say M\, . . . , Mi (derived from Fi, . . . , F^, such that any M can be expressed as where the/'s are forms in xi, . . . , x„_i. Then t = 1 F = FoP'-{- ^fiF^+R', P'^P-Zf^P^, R'^N-Xf,Ni. 1=1 Each exponent of Xn in R' is ^r — 2. We segregate its terms in which this exponent is exactly r — 2 and have F = FoP'-i- i JiF.-hM'xZ-'^+N', «=i where M' is a form in xi, . . . , Xn-i, and N' a form in Xi, . . . , x„ whose order in x„ is ^ r— 3. The theorem is apphcable to the set of forms M\ so that each is a linear combination of M\, . . . , M'm, corresponding to i^'^+i, . . . , Fi+jn, say. As before, F differs from a linear combination of Fq, . . . , Fi+mhy M"x/-^-\-N", where M" is a form in xi, . . . , Xn-i and TV" is a form whose order in Xn is ^ r — 4. Proceeding in this manner, we see that F differs from a linear combination of Fq, . . . , Ft by a form R in xi, . . . , x„^i. One more step leads to the theorem. 49. Finiteness of a Fundamental System of Invariants. Con- sider the set of all invariants of the binary forms /i, . . . , fa, 74 ALGEBRAIC INVARIANTS homogeneous in the coefi&cients of each form separately. By the preceding theorem, there is a finite number of these invariants /i, . . . , Im in terms of which any one of the invariants / is expressible linearly: (1) / = £i/i + . . .-\-EmIn., where Ej is not necessarily an invariant, but is a polynomial homogeneous in the coefficients of each fi separately. Let ai, a2, ... be the coefficients in any order of/i, . . . ,/d. Let A I, A2, . . . be the coefficients in the same order of the forms obtained from them by applying a linear transformation of determinant (l??). We may write I{A) = {^r,)^I{a), Ij{A) = avrjTj{a), EAA)=Gj, where Gj is a function of the a's, |'s, 77's. From the identity (1) in the a's, we obtain an identity by replacing the a's by the A's. Hence m ihYl^ ^Gji^vYJlj, in which the arguments of the /'s are a's. Thus Gj is of order X — X; in ^1, I2 and of order X — X^ in rji, 772. Operate on each member by V^. By § 43, the left member becomes (X + 1)(X!)2/. By the formula to be proved in § 50, the right member becomes :^IACo(^r,)'^-^Gj-^C,{h)^^-^+'VGj-h. . .+a{^vY^V'Gj\, i = i where the C's are numerical constants. Since Gj is of order v = X — X^ ^ in ^1, ^2 and of order p in 771, 772, V+'Gj^O, V^+-Gj = 0, ..., V^Gj = 0. Also Co, Ci, . . . , Cy-i are zero since they multiply powers of (^rj) whose exponents —v, —v-\-l, . . ., X^ — X+j^— 1= — 1 are negative. Hence m (X+l)(X!)2/= ZljC^VGj. §50] FINITENESS OF COVARIANTS 75 The torm obtained from/t = Q;2" by our linear transformation has the coefficients (1), § 45. The polynomial G] in these coefficients is therefore a sum of terms each a product of a constant by v factors of type a^ and v factors of type or,. Hence, by § 44, VCj is a polynomial in the determinantal factors (a/S) and is consequently an invariant of the forms fi. Thus m 1= S//0, where I'j is an invariant. Then, by (l), TO m k=\ i,k=\ By repeating the former process on this /, we get m j,k=\ where the /" are invariants of the forms ft. Since there is a reduction of degree at each step, we ultimately obtain an expression for / as a polynomial m Ii, . . . , In with numerical coefficients. 50. Lemma. If D= ^ir]2— ^2Vi, cind P is homogeneous {of order X) in |i, ^2, and homogeneous {of order n) in t/i, 772, then m (1) F"»Z)"P= 2 GD" -'"+'■ FP, r=0 where Co, . . . , Cm are constants. First, we have FZ)P = P + r/2— +^1— +/>- ^^^ 'dri2 dh dhdm -{-P-h^-ml-+D:^)=-2+X+^)P+DVP, \ dh 9171 9^29771/ by Euler's theorem for homogeneous functions (§24). If P is replaced by D"-^P, so that X and m are increased by n — 1, we get VD"P = {\-\-fx-\-2n)D''-^P-\-DVD''-'^P. 76 ALGEBRAIC INVARIANTS Using this as a recursion formula, we get which reduces to the result in §43 if P = l, whence X = /i = 0. Hence (1) holds when m = l. To proceed by induction from m to w + 1, apply V to (1). Thus m r=0 In the result for VD"'P, replace n by n—m-\-r and P by V^P, and therefore diminish X and m by r. We get Yf£)n-m+rYrp\ _^ £)n -m+r -lyrp \£)n—m+rYr + lp where tr={n—m-\-r){\-\-ij. — r-{-n—m-\-l). Hence, changing r+l to r in the second summand, we get m+l ym+lj)np^ V (Crtr + Cr-l)D''-'^+'-^V'P, r=0 with C„,+i = 0, C_i = 0. Thus (1) is true for every m. 51. Finiteness of Syzygies. Let /i, . . . , 7^ be a funda- mental system of invariants of the binary forms /i, . . . , fa- Let S{zi, . . . , Zm) he a. polynomial with numerical coefficients such that S(Ii, . . . , /w),Vhen expressed as a function of the coeflficients c of the /'s, is identically zero in the c's. Then 5(7) = is a syzygy between the invariants. By means of a new variable z^+i, construct the homogeneous form 5'(zi, . . ., z^+i) corresponding to S. By §48, the forms S' are expressible linearly in terms of a finite number S\, . . . , yt of them. Take 2^+1 = L Thus (1) 5 = Ci5i + . . .+CA where Ci, . . . , Ct are polynomials in zi, . . . , z„,. Take zi =Ii, . . . , Zm = Im- Hence there is a finite number of syzygies Si=0, . . ., St = 0, such that any syzygy 5 = implies a relation (l) in which Ci, . . . , C* are invariants. In particular, every syzygy is a consequence of 5i = 0, . . . , 5t = 0. § 52] TRANSVECTANTS 77 52. Trans vectants. Any two binary forms have the covariant (1) (/, 0)'- = (a^)'-«/-'-/3i-'-, called the rth. transvectant (Ueberschiebung) of / and <^, and due to Caj'ley. It is their product if r = 0, their Jacobian if r = l, and their Hessian \i J=<t> and r = 2, provided numerical factors are ignored (Exs. 4, 5, § 40). It may be obtained by differentiation and without the use of the symbolic notation. In fact, a special case of (l), § 44, is so that if/ is of order k and 4> of order /, (2) im, 0(^))^ = fc^' ^^[F'-/(^)c/>fr,)l, = ,. x\ftcr/(ti, ^2)-4>(m, m) is operated on by V, we set 771 = ^1, For example, let /(^)=a^a^, </.($) = 7^\ P^a^S^y '. Then d^idvi ^ * 9^29771 * * The difference is VP. Taking r]i= ^i, 772= I2, we get The numerical factor in (2) is here I/6. Hence (3) ^<^j y,')' = h{0yh^yf+K'xy)0^y^'. In general, consider the two forms Then by (l), § 44, and the Corollary, and by (2), where the summation extends over all the combinations of the 78 ALGEBRAIC INVARIANTS as r at a time, and over all the permutations of the /S's r at a time. Thus the number of terms in the sum is the reciprocal of the factor preceding S. If the a's are identified and also the /3's, (4) becomes (1). li k = 2, l=S, r=l, we have one-sixth of a sum of six terms; then if the /3's are identified we have two sets of three equal terms and obtain (3). Since F is a differential operator, (2) gives (5) (2c/,, i:kj<f>jy = ^^cMft, 4>,)r. Apolarity; Rational Curves, §§ 53-57 53. Binary Forms Apolar to a Given Form. Two binary quadratic forms are called apolar if their lineo-linear invariant is zero; then they are harmonic (Ex. 3, § 11). In general, the binary forms i=Q\l/ i=o\t/ of the same order, are called apolar if (1) {a^y'^=lji-i)'(jyA.-i=o. In particular, / is apolar to itself if n is odd (Ex. 4, § 38). Let the actual linear factors of <^ be /3x^^\ • • • , /3j'^"^ By (1), (4), § 52, (a^)" = (a/, /3,(i> . . . /3,("))" = (a|3(i)) . . . (a/3("^). But ^x^"^ vanishes if xi and X2 equal respectively Thus Hence if vanishes for xi=yi''''\ X2=y2^''^ (r = l, . . . , n), it is apolar to f if and only if Thus / is apolar to an actual wth power (y2A;i— ^1^2)" if and only if q:j,'* = 0, i.e., if yi, y2 is a pair of values for which /=0. § 53] APOLARITY 79 If no two of the actual linear factors /< of / are propor- tional, / is apolar to n actual «th powers //' and these are readily seen to be linearly independent. Then their linear combinations give all the forms apolar to/. For, if/ is apolar to <^i, . . . , 0n, it is apolar to /^i<^i + . . •-\-kn4>n^ where ^i, , . . , ^a are con- stants, since, by (5), § 52, (/, y^i0i+. . .+K^nY = k,{J,<j>iY+. . .-\-knU,4>nY = o. Moreover,/ is not apolar to n-\-\ linearly independent forms 01, 02, • • . , 0n+l- For, if so, we have n-\-l equations like (1), in which the deter- minant of the coefficients of ao, • • • , an is therefore zero. But this implies a linear relation between the 0's. /// is the product of n distinct linear factors U, a form can be repre- sented as a linear combination of /i", . . . , In^ if afid only if is apolar to f. In particular, if r and 5 are the distinct roots of f^ax'-\-2bx-\-c = 0, the only quadratics harmonic to / are g{x-rY+}i{x-sY. In case h, . . . , Ir are identical, while li^lt{i>r), we may replace /i", . . . , /r" in the above discussion by /i", /i"~^X, . . . , l^n-r+iy-i^ whcrc X is any linear function of Xi and x-y which is hnearly independent of h. In fact, after a linear trans- formation of variables, we may set li=X2, \ = Xi. Then the above r forms have the factor 0:2""''"^^ and hence are of type with bi = 0{i ^ n—r). Also, / now has the factor x-z^, so that ai = 0{i<r). Hence every term of (l) is zero. For example, /=.Vi-.r-..(.Vi — .Vi)- is apolar to Xi^, Xi^Vi] .r2»; (xi — x-2)\ (.fi — .^2)^^:1, which give five linearly independent quintics. In general, when there are multiple factors of /, the n forms apolar to / obtained above can be proved to be linearly independent. This fact is not presupposed in what follows. 54. Binary Forms Apolar to Several Given Forms. From the list of the given forms we may drop any one linearly de- 80 ALGEBRAIC INVARIANTS pendent on the others, since a form apolar to several forms is apolar to any Hnear combination of them. In the resulting linearly independent forms the g-rowed determinants in the rectangular array of the coefficients are not all zero. For, if so, there are solutions ^1, . . . T kg, not all zero, of kiaii-\-k2aio-\- . . .-\-kgaig = {i = 0,l,. . . ,n), which would give, contrary to hypothesis, the identity If io^i" + . . . is apolar to each/r, then 2(-l)Y".)aJ;„_,- = (r = l,. . . , g). These determine g of the b's as linear functions of the remaining 5's, which are arbitrary. Hence there are exactly n-\-l—g linearly independent forms apolar to each of the g given linearly independent forms. In particular, apart from a constant factor, there is a single form apolar to each of n given linearly independent forms of order n. Consider three binary cubic forms /i =aJ^=aoXi^+3aixrx2-}-^a2XiX2^-\-a3X2^, J2 = /Sx^ = h(iXi^ -\-2,hxXi^X2-\-^h2XiX2'^ +h2,X2^ , Jz=-'yx^^CQXi^-\-ZCiXi~X2+2>C2XiX2^-{-C2X2^. Each is apolar to the cubic form = (a/3)(a7)(i37)«:ri3^Tx. For, by (4), § 52, and the removal of a constant factor by (5), (<^, 5.-^)3 = («i3)(a7)(/3T)(«5)(/35)(75), which is changed in sign if 6 is interchanged with a, /3, or 7, §54] APOLARITY 81 and hence is zero if bj^ is one of the fi. Hence each /< is apolar to 4). Now ai" a\a.2 of^ 71^ 7172 72" In fact, the determinant vanishes if (ajS) = as may be seen by setting ^i=cai, ^2=ca2. Moreover, the two members are of total degree six and the diagonal term of the determinant equals the product of the first terms 0:1182, etc., on the left. Since ai^ax=ai^xi-i-ai^a2X2=aoXi-{-aiX2, etc., we find, by multiplying the members of the last equation by ax/3x7z, aoXi-j-aiX2 ai:ri+a2^2 02^1+03X2 boXi-\-biX2 biXi-{-b2X2 b2Xi-\-b3X2 CoXi-\-CiX2 ^1:^1+^2^:2 C2X1+C3X2 where = [012]xl3 + [013]A-l%2 + [023]xlX22 + [123]:^;2^ at Qj at [ijk] = 1 bi bj bt Ci Cj Ct If (p is identically zero, the four three-rowed determinants in the rectangular array of the coefficients of /i, /2, /a are all zero, and the/'s are linearly dependent. Apart from a constant factor, is the unique form apolcr to three linearly independent cubic forms fi, f 2, fz- The extension to n binary w-ics is readily made. 55. Rational Plane Cubic Curves. The homogeneous coor- dinates ^, rj, f of a point on such a curve are cubic functions of a parameter /. We may take / = a:i/.T2 and write Ps=/l, PV^f2, P^^fd, where p is a factor of proportionality and the /'s are the cubic forms in § 54. We may assume that the/'s are linearly independent, since otherwise all of the points (^, tj, f) would He on a straight line. 82 ALGEBRAIC INVARIANTS There is a unique cubic form ^ apolar to /i, /2, /a (§ 54). This cubic form, denoted by (p = (p/, is fundamental in the theory of the cubic curve. Three points determined by the pairs of parameters xi, X2; yi, y2', and z\, so, are collinear if and only if (1) 4>x4>Az = ^- For, if the three points he on the straight Hne (2) /t + ;«r7 + «r = 0, the three pairs of parameters are pairs of values for which (3) C{xi, x-f) ^lfi-^mf2-{-nf3 = 0. Since C is apolar to (p, (1) follows from the first italicized theorem in § 53. Conversely, (1) implies that the cubic C which van- ishes for the three pairs of parameters is apolar to ^ and hence (§ 53) is a linear combination of /i, /2, /s, say (3); the corre- sponding three points lie on the straight line (2). Since (2) meets the curve in three points the ratios xi/x-z of whose parameters are the roots of (3), the curve is of the third order. We restrict attention to the case in which the actual linear factors ax, ^x, ix of 4> are distinct. Since any cubic apolar to </> is a linear combination of their cubes (§ 53), /i = rna.3+Ct2/3/+c,37/ (^ = 1, 2, 3). Since the determinant \ dj \ is not zero, suitable linear com- binations of the/'s give a/, /3x^, jx^. Hence by a linear trans- formation on ^, 7], f (i. e., by choice of a new triangle of ref- erence), we may take * The line ^ = is an inflexion tangent, likewise 7j = and f = 0. In addition to the resulting three inflexion points, there are no others. For, at an inflexion point three consecutive points are collinear, so that (l) gives (^ = <^/ = 0. In the present * We now have the formulas in the second part of § 54, where now ax^ is the actual, not a symbolic, expression of /i, etc. §56] RATIONAL PLANE CUBIC CURVE 83 case there are therefore exactly three inflexion points and they are coHinear. 56. Any Rational Plane Cubic Curve has a Double Point. Let Px denote the point (^, r/, i") determined by the pair of parameters xi, xo. If the ratios x\/x2 and yi/y2 are distinct and yet Pi coincides with Py, then Px is a double point. For, any straight line (2), § 55, through Px meets the curve in only the three points whose pairs of parameters satisfy the cubic equation (3), and since two of these pairs give the same point Px. the line meets the curve in a single further point. Hence there is a double point Px = Py if and only if there are two distinct ratios xi/xo and yi/>'2 such that (l) holds identically in Si, 22- Let Q be the quadratic form which vanishes for the pairs of parameters xi, X2 and yi, yo giving a double point. By (1), and the first theorem in § 53, Q is apolar to <f)x~(f>z for Si, Z2 arbitrary. Write (f)'i^ as a symbolic notation for <f>, alter- native to ^/. Applying the argument made in § 54 for three cubics to two quadratics, we see that the unique quadratic (apart from a constant factor) which is apolar to both <^^-0, and (t>'x'^4>'u> is their Jacobian /= (<^0 )<t>x4> X* 020 W Since and 0' are equivalent symbols, their interchange must leave / unaltered. Hence -^ = 2(00 )0x0 x{(t>z4>'w — 4>'z4>tc\. The quantity in brackets equals {<i><i>'){zw) by (l), § 40. Dis- carding the constant factor \{zw), we may take as the desired quadratic form. This is the Hessian of 0. Conversely, the pairs of values for which Q vanishes are the pairs of parameters of the unique double point of the curve. 57. Rational Space Quartic Curve. Such a curve is given by 84 ALGEBRAIC INVARIANTS where the four binary quartics are linearly independent. By § 54, there is a unique quartic apolar to each of the four. As in § 55, four points Px, Py, Pz, Pw on the curve are coplanar if and only if Thus = gives the four points at which the osculating plane meets the curve in four consecutive points. It may be shown that the values i"/*\ X2^'^^ for which the Hessian of </> vanishes give the four points i'j:^') on the curve the tangents at which meet the curve again. Fundamental Systems of Covariants of Binary Forms §§ 58-63 58. Linear Forms. A linear form otx is its own symbolic representation. If ax = 0x, then (q:/3)=0. Hence the only covariants of ax are products of its powers by constants. A fundamental system of covariants of n linear forms is evidently given by the forms and the ln(n — l) invariants of type (a/3), where ax and 0x are two of the forms. 59. Quadratic Form. A covariant K of a single quadratic may have no factor of type (a/S) and then it is ax Px Ifx • . • J , or may have the factor (a/S) and hence the further factor {a0), {ay){p8), {ay)^x, or axl3x, including the possibility 5 = 7. In the first case, K^{a^)^Ki, where A'l is a covariant to which the same argument may be applied. Now {a'y)^ay if ^1=72, y2=—yx. Hence in the last three cases, K has a factor of the type e = {a^)ay^^, where ay is either ax or a new mode of writing (0:7), and similarly jSj is either ^x or a new mode of writing (/35). Interchanging the equivalent symbols a and 8, we get d - i^a)^,a, = K«/3) (a,/3, - /3,«.) = § (a^)2(y3), §eO] FUNDAMENTAL SYSTEM OF COVARIANTS 85 by (1), § 40. We are thus led to the first case. Hence the fundamental system of covariants of / is composed of / and its discriminant. EXERCISES 1. The fundamental system for f=ax^ = bx- and l=ax = 0x is /, /, {ab)*, {aocy, {aa)ax. 2. The fundamental system for f=ax^ = bx- and <t> = ax^ = Px^ is/, <t>, {aby, (a^)^, (oa)-, {aa)axax. Hint: {act) {a0)aaliy = {aa) '-fiy^z-\{a0) ^GyOz, as proved by multiplying together the identities (Ex. 6, § 40) {a^)ay = {ap)ay-{aa)0^, (a/3)(Zz=(d/3)as-(aa)/5,, and noting that a and /3 are equivalent symbols. 60. Theorems on Trans vectants. In the expression (4), § 52, for a transvectant, each summand taken without the prefixed numerical factor is called a term of the transvectant. In the first transvectant (3), § 52, the difference of the two terms is by Ex. 6, § 40, and is the negative of the 0th transvectant (viz., product) of (a/3) and y^^. The act of remo\dng a factor a^ and a factor jS^ from a product and multiplying by the factor (q:/3) is called a convolution (Faltung). We have therefore an illustration of the following Lemma. The difference between any two terms of a trans- vectant equals a sum of terms each a term of a lower transvectant of forms obtained by convolution* from the two given forms. Consider the rth transvectant of where P and Q are products of determinantal factors. Then PQ is a factor of each term of the transvectant. Any two terms T and T differ only as to the arrangements of the as and the /3's. Hence T' can be derived from 7 by a permuta- * Including the case of no convolution, as 7^' from itself, in the above example. 86 ALGEBRAIC INVARIANTS tion on the as and one on the /S's, and hence by successive interchanges of two a's and successive interchanges of two jS's. Any such interchange is said to replace a term by an adjacent term. For example, the two terms of (3), § 52, are adjacent, each being derived from the other by the inter- change of a with /3. Between T and T' we may therefore insert terms Ti, . . . , Tn such that any term of the series T, Ti, T2, ■ ■ ' , Tn, T' is adjacent to the one on either side of it. Since r-r=(r-ri)+(ri-r2)+. . .+(r„_i-r„)+rr„-r), it suffices to prove the lemma for adjacent terms. The interchange of two a's or two /3's affects just two factors of a term of (4), § 52. The types of adjacent terms are * where /3' and /3" were interchanged. The difference of the last two terms is seen to equal C(jS"^')a\ by the usual identity. The latter is evidently a term of the (r — l)th transvectant of / and {l3"^')4>/{^'\^\], which is obtained from <f) by one convolution. The difference of the first two adjacent terms equals C(aW')(fi'^"), since a la I p i p 1 {a'a")(p'n - ia'^'){a"n + icc'fi"){a"^') ^\ I n n> o" a 2OL 2P2P 2 ' // Qt o" a \a i p 1 p 1 a 20i 2 P 2 p 2 = 0, as shown by Laplace's development. The same relation follows also from the identity just used by taking |i = — a"2, ^2= a" I. The resulting difference is a term of the (r-2)th transvectant of a ^a ^ p iP ^ which are derived from / and by a convolution. *A pair C(a'0')a"^, C(a"li')a'^, obtained by interchanging a and a", is essentially of the second type. §G0] FUNDAMENTAL SYSTEM OF COVARIANTS 87 The LcMnma leads to a more important result. By tlie proof leading to (4), § 52. the coefficient of each term of a transvectant is 1/N, if .Y is the number of terms. Just as S = hiTi-\-T-2) implies S-Ti=h(T2-Ti), so impUes S = ^{Ti + . . . + r.v) Hence ilic clijjcrciicc bclwccii a traiisvccia>i! a)iJ any one of its terms equal a sum of lernis eaeJi a term of a lower iransveeiant of forms obtained by eonvolntion from the two givefi forms. Each term of a lower transvectant may be expressed, by the same theorem, as the sum of that transvectant and terms of still lower transvectants, etc. Finally, when we reach a 0th transvectant, i.e., the product of the two forms, the only term is that product. Hence we have the fundamental Theorem. The differ enee between any transveetant and any one of its terms is a linear function of lower Iransvcclanls of forms obtained by convolution from the two given forms. For example, from (3), §52, and the result preceding the Lemma, we have and {ajS) is dcri\oil from (Vcp\ by one convolution. 61. Irreducible Covariants of Degree m Found by Induction. Let / = a/ = /3V' = . . .=X/ be the binary ;;-ic whose fundamental s}'stem of covariants is desired. Since a term with the factor (a/3) is of degree at least two in the coefficients of /, the only covariants of degree unity are kf, where k is a numerical constant. We shall say that / is the only irreducible covariant of degree unity, and that/, A'l, .... /v, form a complete set of irreducible covariants of degrees <m if every covariant of degree <;;/ is a poly- 88 ALGEBRAIC INVARIANTS nomial in f, ..., Ks with numerical coefficients. Given the latter, we seek the irreducible co variants of degree m. A covariant of degree m is a polynomial in the (a/S) and the ax such that each term contains m letters a, ^, y, . . . . Let Tm be one of the terms with its numerical factor suppressed. Let a, /3, . . . , K, X be the m letters occurring in Tm, so that r„ = P(aX)«(/3X)^ . . . {kXYxJ {a-\-b + . . .+k-^l = n), where P involves only a, /3, . . . , k. Then r^_i=Pa.«/3/ . . . K,fc is a covariant of degree m—l. Evidently Tm is a term of {Tm-i, X/)^ {r = n-l), since it is obtained by r = a-\-b^. . .-{-k convolutions from Tm-i^x"^- By the final theorem in § 60, Tm={Tm-ljy+^^'cj{frn-ufy, j=0 where the Cj are numerical constants, and each Tj^-i is derived from r„i_i by convolutions and hence is a covariant of degree m—l. But the covariant of degree m was a linear function of the various Tm- Hence every covariant of degree m oj f is a linear function of transvectants {Cm-ufY of covariants C^-i of order m—l with f. Such a transvectant is zero if k>n, in view of the order of/. Moreover, it suffices by (5), §52, to employ the C^-i which are products of powers oif,Ki,. . . , Kg. Hence the covariants of degree m are linear functions of a finite number of transvectants. In the examination of these transvectants {Cm-i, fY, we first consider those with k = \, then those with k = 2, etc. We may discard any {Cm-i,fY for which Cm-i has a factor <t>, of order '^k, which is a product of powers of f, Ki, . . . , Ks, and of degree <m-l. For, if T is a term of (c^,/)^, and if C^_i=^0, then T is obtained by k convolutions of 0/, and qT by the same k convolutions of g<^/, not affecting q. Hence qT is a term of {q(t>, fY' Hence (Cm-i, fY-qT+~^' Cj(Cm-i,fy. 3=0 § G2] FUNDAMENTAL SYSTEM OF CO VARIANTS 80 But the terms of the last sum have by hypothesis been con- sidered previously, while the co variants ^ and T are of degree * <m and hence are expressible in terms of/, Ki, . . . , Ks. 62. Binary Cubic Foim. The only irreducible co variant of degree one of was shown to be /. The only covariants of degree two are (a^)'-a/-%-'^-'- (r = 0, 1, 2,3). This vanishes identically if r is odd. If r = 0, we have /-, which is reducible. Hence the only irreducible covariant of degree two is ia0)-a,l3, = (/, f)~ = Hessian H of /. To find the irreducible covariants of degree m=S, we have Cjn-i = Ii or/-. In the second case, C^-i has the factor f of degree <m — l and order 3^^' (since we cannot remove by convolution more than three factors from the second function f in the transvectant). Hence we may discard Cm-i=f~- It remains to consider (H, f)^\ k^l, 2. Now (H, /) = (a/3)2(a7)/3^7/ = Jacobian / of // and / is irreducible, being of order and degree three and hence not a polynomial in / and H. Next, {H, jy - (a/3)2(a7) (^7)7x = P{a^h., P = {cc0) {ay) (fiy) . Interchanging a with 7, we get P{^y)ai. Interchanging jS with 7, we get P(y(x)0j. Hence (F,/P = |F|M)7. + (^7K + (7«)/3x|=0. The irreducible covariants of degree three or less are therefore /, H, J. To find those of degree 7;z=4, we have Cm-i=P, fB, J, * This is evident for the factory of Cm-i- Since <f) is of degree <w— 1, the term T of (<^, /)^' involves fewer than m — 1 + 1 symbols a, ^, . . . , and hence is of degree < m. 90 ALGEBRAIC INVARIANTS of which the first two may be discarded as before. It remains to consider (/,/)*, for ^ = 1, 2, 3. By § 52, = (a/3)2(a7) lKi35)7x-5.2 + |(y5)i3.7x5/|. Replacing {P8)yz by iy8)^s + il3y)8„ and noting that {amay){^yH5.' = {H,f)--f=0, we get (/,/) = (a/3)2(a7)(75)/^.7x5/. Interchange y and 5, Hence U,/)=K«/3)'(75))3.7x5xK«7)5x + (5a)7x}. The quantity in brackets equals —{yd)ax. Hence Denoting H by Jh^ = h"^i, we have / = (//,2^^^3) = (/,^)/,^^^2^ / = ^: X 7 by the theorem in § 60. Here J = {haYax = {H, f)^ = 0. Since the first term is changed in sign when a and /3 are interchanged, we have {J,fy = 0. For the third case, (/, /)3 = (M)2(a7)^x7x2, 5.3)3 = {amay)(^8){y8y = D, an invariant, evidently equal to {H, H)^, the discriminant of H. Thus D is the discriminant of / (§§8, 30) and is not identically zero. Hence D is the only irreducible covariant of degree four. We can now prove by induction that /, H, J and D form a complete set of irreducible covariants of degree ^ ;w ^ 5. Let this be true for covariants Cm-\ of degree ^w — 1. We may discard (C^-i, fy if Cm-i has the factor/ or /, each of which is of order 3 ^ ^ and of degree (1 or 3) less than m — 1; and evidently also if it has the factor D. Hence Cm-\=n% e ^ 2. If ^ J 2, it has the factor E of order 2^k and degree 2<m — l. It remains to consider {E% f)^. If e>2, H^ has the factor § 61] LAW OF RECIPROCITY 91 H' of order 4^3 and degree -Kw— 1, since H'^ is of degree ^ 6. Finally, {H\ / )3 = {hVl\, a/)3 = {ha)Hh'a)h\ = {h\ (JlaY-a;) = 0. Hence /, H, J, D form a fundamental system of covariants {cf. §30). 63. Higher Binary Forms. The concepts introduced by Gordan in his proof of the liniteness of the fundamental system of covariants of the binary p-ic enabled him to find * the system of 23 forms for the quintic, the system of 2G forms for the sextic, as well as to obtain in a few lines the system for the cubic (§ 62) and the quartic (§ 31). Fundamental sys- tems for the binary forms of orders 7 and 8 have been deter- mined by von Gall.f Gordan's method yields a set of covariants in terms of which all of the covariants are expressible rationally and integrally, but does not show that a smaller set would not serve similarly. The method is supplemented by Cayley's theory | of generating functions, which gives a lower limit to the number of covariants in a fundamental system. 64. Hermite's Law of Reciprocity. This law (§ 27) can be made self-evident by use of the symbolic notation. Let the form <t)=ax^ = ^x^ = - . .=ao{xi — piX2){xi— P2X2) . . . {xi — ppX2) have a covariant of degree d, K = ao'^^ipi-p-2)Kpi-p-syip2-P3)^ . . . (xi-pi-rs)'' . . . (xi-ppa:.)^ so that each of the roots pi, . . . , pp occurs exactly d times in each product. Consider the binary ^-ic J = ai'^ = b/ = . . . = CoCti— riXo) . . . {xi — raXo). * Gordan, Inmriantentheorie, vol. 2 (1887), p. 236, p. 275. Cf. Grace and Young, Algebra of Invariants, 1903, p. 122, p. 128, p. 150. t M alhematische Annaleii, vol. 17 (1880), vol. 31 (18S8). X For an introduction to it, see Elliott, Algebra of Quantics, 1895, p. 165, p. 247. 92 ALGEBRAIC INVARIANTS To the various powers, whose product is any one term of K, {pi-p'zY, {pi-p-j.y, {p2-p-iY, . . ., (X1-P1X2)'', {xi-p2X2)^, . . ., we make correspond the symboUc factors {ahy, {acy, (be)'', . . ., aj\ h\ . . . of the corresponding covariant of/: C={ahy{acy{bcY . . . aj'hjx'j' . . ., of degree p (since there are p symbols a, b, c, . . . , cor- responding to pi, . . . , Pj,) and having the same order /i+^2+/3 + . . . as A'. Conversely, C determines A. EXAMPLES Let p = 2. To K = au^^(pi — Pi)^'' corresponds the invariant C= (ai)^^ of degree 2 of f=(ix'^ = hx'^^. Again, to the covariant K4>^ of (/> corresponds the covariant (ab)'^^ OxVV of the form ax'^'^'^^ = bx^^'^K Concomitants of Ternary Forms in Symbolic Notation, §§ 65-67 65. Ternary Form in Symbolic Notation. The general ternary form is '' r\s\l\ where the summation extends over all sets of integers r, 5, /, each = 0, for which r-\-s-\-l = n. We represent/ symbohcally by / = a_," = |3/ . . . , ax = a\X\-^a2X2-\-OL2,Xz, .... Only polynomials in ai, 02, "a of total degree n have an inter- pretation and Just as a\^2—(X2^\ was denoted by (a/S) in §39, we now write (a/37) a\ 0:2 OL-i ^1 /32 /33 71 72 73 §66] CONCOMITANTS OF TERNARY FORMS 93 Under any ternary linear transformation T: ^i=^tXi + 7jtX2 + f«X3 (j = l,2, 3) ax becomes Q:{Xi+a,X2+«5-A'3, and/ becomes fi ! Z , '. ATstX^\X''2X^-s = (aiXi-\-a^X2-\-oiiX:i) . risltl Thus ax behaves Uke a covariant of index zero of /. Also «{ a, «{• ^j /3, ^f =M7J(^r7f), 7£ 7., 7r so that (aj37) behaves like an invariant of index unity of/. EXERCISES 1. The discriminant of a ternary quadratic form ax^ is ^ {aftyY. 2. The Jacobian of a^', Px"^, tz" is /ww (a/3TJa:c^~Vi'"~V/~^ 3. The Hessian of a^" is the product of (a/37)2a/~^/3z"~^7i"~^ by a constant. 4. A ternary cubic form ai' = /3i' = . . . has the invariants {a0y){a0b){ayb){fiyb), (a0y)(a0o)(aye)(0y<}>)(oe<t>y. 66. Concomitants of Ternary Forms. If ui, ii2, uz are constants, Uz = UiXi-\-U2X2+U:iX3 = represents a straight line in the point-coordinates xi, xo, X3. Since ui, U2, M3 determine this line, they are called its line- coordinates. If we give fixed values to xi, X2, X3 and let the line-coordinates ui, U2, ws take all sets of values for which Ux = 0, we obtain an infinite set of straight lines through the point (xi, X2, X3). Thus, for fixed x's, Ux = is the equation of the point (xi, X2, x^) in line-coordinates. Under the linear transformation T, of § Go, whose deter- minant {^vO is not zero, the line Ux = is replaced by Ux=UiXi-{-U2X2 + U3X3=0, in which 333 f/i = S ^iUi, 1/2= ^ ViUi, t/3 = - fi«<- t=l i=l 1=1 94 ALGEBRAIC INVARIANTS The equations obtained by solving these define a linear trans- formation Ti which expresses wi, U2, us as linear functions of Ui, U2, Us and which is uniquely determined * by the transformation T. Two sets of variables Xi, X2, X3 and wi, ti2, U3, transformed in this manner, are called contragredient. A polynomial P{c, x, u) in the two sets of contragredient variables and the coefficients c of certain forms Ji{x\, X2, xz) is called a mixed concomitant of index \ of the /'s if, for every linear transformation T of determinant A?^0 on xi, X2, X3 and the above defined transformation Ti on ui, U2, uz, the product of P{c, X, u) by a'' equals the same polynomial P(C, A', U) in the new variables and coefficients C of the forms derived from the /'s by the first transformation. For example, Ux is a concomitant of index zero of any set of forms. In particular, if P does not involve the w's, it is a covariant (or invariant) of the /'s. If it involves the u's, but not the re's, it is called a contravariant of the /'s. Since U\=u^, U2 = u^, 1/3=11^, we see by the last formula in § 65, with 7 replaced by u, that (a^u) behaves like a contra- variant of index unity of ax^, and also like one of ax", ^x^. For the linear forms ax and fix, («/3«) has an actual interpretation. For /=«!= = /So:-, where /= a2ooXr +ao2oX2^ +0002X3^ +2auoXiX2 +20101X1X3 +2aonX2X3, it may be shown that O200 duo ClOl 111 Olio flo20 floil Ui Gioi flon aoo2 ^h III Ui Hi {afiuY By equating to zero this determinant (the bordered discriminant of /), we obtain the line equation of the conic /= 0. 67. Theorem. Every concomitant oj a system of ternary forms is a polynomial in Ux and expressions of the types ax,{a^y), {a^u). * We have only to interchange the rows and columns in the matrix of T and then take the inverse of the new matrix to obtain the matrix of the transforma- tion Ti. Similarly, Xu x-i are contragredient with Ui, Ui, if we have T, § 40, and «i = (^72 Ui - & Ui) / (It?) , Ui = ( - ,,1 t/i+ ti Ui) I ( i-n) . 167] CONCOMITANTS OF TERNARY FORMS 95 a a a ^2 a a ^3 a _a_ dm _a_ a^ 9171 a^ia'72ar3 a afi a af2 a ars A concomitant of the forms /<(xi, X2, X3) is evidently a covariant of the enlarged system of forms /i and Ux. We may therefore restrict attention to covariants. In the proof of the corresponding theorem for binary forms, we used the operator (1), § 42. Here we employ an operator V composed of six terms each a partial differentiation of the third order: V the determinant being s>Tnbolic. It may be shown as in § 43 that F(^77f)" = «(» + l)(w + 2)(|r?f)"-l. As in § 44, the result of applying F'" to a product of k factors of the type «{, / factors of the type /3„ and m factors of the type 7J-, is a sum of terms each containing k—r factors a^, l—r factors /3„ m—r factors 75-, and r factors of the type (a/37). For the case of an invariant /, the theorem can be proved without a device. In the notations of § 65, we have /U) = (^,f)V(a). Each yl is a product of factors aj, a,, a^. Hence I {A) equals a sum of terms each with X factors of the type a^, X of t}'pe a„ and X of type a^. Operate on each member of the equation with F^. The left member becomes a sum of terms each a product of a constant and factors of t}^e (ajSy). The right member becomes the product of 7(a) by a number not zero. Hence I equals a polynomial in the (a^y). For a covariant K, we have, by definition, " K(A,X)=(^rji)^K{a,x). Solving the equations of our transformation T in § 65, we get (|77i')A'i =a;i(7?2S''3 — ^3^2) +^"2(773^1 — •71^3) -}-X3{r]i^2 — 772ri), 96 ALGEBRAIC INVARIANTS etc. Replacing xi by y2Z3-ys^2, X2 by yaZi-yiSs, and xs by yiZ2—y2Zi, we get {^vOXi^yr,z^-y^Zr„ {h^)X2 = ycZ^-yiZ^, (^v^)X3=yiZ,-y„Zi. Our relation for a covariant K of order co now becomes S(product of factors a^, y^, z^, a,,, . . . , z^) = {^r]^Y'^''K{a, x) . each term on the left having \+co factors with the subscript ^, etc. Apply the operator V to the left member. We obtain a sum of terms with one determinantal factor (a/37), ("i3y) or {ayz)=ax, and with X+co — 1 factors with the subscript |, etc. The result may be modified so that the undesired factor (a^y) shall not occur. For, it must have arisen by applying V to a term with a factor Hke a^^^y^ and hence (by the formulas for the Xi) with a further factor s, or z^. Consider therefore the term Ca^jS^y^z, in the initial result. Then the term — Caj|8,>\Sf must occur. By operating on these with F, we get C{a^y)Zr„ — C(a/3s)y„ respectively, whose sum equals CI (/3y2K - (ays)^,N C(/3.a, -ax/3,) , as shown by expanding, according to the elements of the last row. Oil /3i yi zi a2 /32 )'2 Z2 «3 /33 ys Z3 Oir, ^„ ytl Zr, :0. The modified result is therefore a sum of terms each with one factor of type (ai37) or ax and with X+oj — 1 factors with subscript ^, etc. Applying V in succession X + co times and modifying the result at each step as before, we obtain as a new left member a sum of terms each with \-\-w factors of the types {al3y) and ax only. From the right member we obtain nK, where n is a number ?^0. Hence the theorem is proved. §68] QUATERNARY FORMS 97 68. Quaternary Forms. For ax=Q;i:ri+. . .-{-a^Xi, { „ n an », n tn J —OLx = Pi — 7x — Ox has the determinant {a^yb) of order 4 as a symbolic invariant of index unity. Any invariant of / can be expressed as a polynomial in such determinantal factors; any co variant as a polynomial in them and factors of type ax. In the equation «x = of a plane, n\, . . ., W4 are called plane-coordinates. The mLxed concomitants defined as in § 66 are expressible in terms of Ux and factors hke ax, (a^y d) , ia^yu) . For geometrical reasons, we extend that definition of mked concomitants to polynomials P{c, x, u, v), where Vi, . . . , 1)4 as well as wi, . . . , W4 are contragredient to xi, . • . , X4.' There may now occur the additional type of factor {a^Uv) = {ai^2 -Ot2^l) (usVi-UiV^) +. . . -f- (0:3/34 -0:4183) {uiV2 -U2V1). These six combinations of the w's and v's are called the line- coordinates of the intersection of the planes Wx = 0, Vx = 0. For instance, {a^uv)~ = is the condition that this line of inter- section shall touch the quadric surface ax^ = 0. We have not considered concomitants involving also a third set of variables wi, . . • , w^, contragredient with the x's. For, in Mi:»:i-|-. . .-\-u^Xi = 0, z'lXi-f-. . .-\-V4X4, = 0, WiXi-}-. . .+^4^4 = 0, xi, . . . , Xi are proportional to the three-rowed determinants of the matrix of coefficients, so that (auvw) is essentially a*. INDEX (The numbers refer to pages) Absolute invariant, 51 Alternants, 41 Annihilators, 34, 39, 72 Apolarity, 78-84 Binary form, 14, 91 Canonical form of cubic, 17 quartic, 50 ternary cubic, 28 Concomitants, 93, 97 Conic, 2, 21, 24, 94 Contragredient, 94 Contravariant, 94 Convolution, 85 Covariant, 12, 15, 66 — in terms of roots, 56 symbolic factors, 67, 95 — as invariant, 71 Cross-ratio, 5, 15, 56 Cubic curves, 25-29, 81 — form, 14, 16, 48, 80, 89, 93 Degree, 30 Differential operators, 36, 59, 95 Discriminant of binary cubic, 17, 36 quadratic, 10 p-\c, 55 ternary quadratic, 24 Double point, 83 Euler's theorem, 15, 41 Finiteness of covariants, 70-76 syzygies, 76 Forms, 14 Functional determinant, 12 Fundamental system, 48, 61, 84-91 Harmonic, 15, 20, 78 Hermite's law of reciprocity, 45, 91 Hessian, 11, 15-18, 23-28, 58, 66, 84, 93 — curve, 25 Hubert's theorem, 72 Homogeneity, 14, 30, 37 Homogeneous coordinates, 8, 20 Identity transformation, 33 Index, 10, 14, 15, 31, 32 Inflexion point, 26-28, 82 — tangent, 26, 82 — triangle, 27 Intermediate invariant, 19 Interpretation of invariants, 2, 10, 23 Invariant. 1, 10, 14, 28 — in terms of roots, 54 Inverse transformation, 33 Irrational invariant, 55 Irreducible covariant. 87 Isobaric, 31, 32, 38, 42 Jacobian, 12, 15, 18. 29, 65, 83. 93 Leader of covariant. 40. 43, 58 Line coordinates. 93, 97 — equation of conic. 94 Linear form, 9, 14, 33. 84 — fractional transformation, 6, 22 — transformation, 3, 9, 22. 33. 34, Mixed concomitant, 94, 97 Order, 14 Partitions. 44, 45 Perspective, 4 99 100 INDEX Plane coordinates, 97 Product of transformations, 33 Projective, 4, 23 — property, 10, 11,23 Projectivity, 5, 6 Quadratic form, 10, 14, 48, 84 Quartic, 14, 36, 49, 83 Quaternary form, 97 Range of points, 4 Rational curves, 81 Reciprocity. See Hermite. Resultant, 10, 18, 19 Seminvariant, 40, 42-50, 64 — in terms of roots, 53 Singular point, 25 Solution of cubic, 17 quartic, 52 Symbolic notation, 63 Syzygy, 49, 50, 76 Ternary form, 14, 24, 25, 92 Transformation. See Linear. Transvectants, 77, 85 Unary form, 14 Weight, 31, 32, 38 Return on or before IMT 01 199&I W2fll,D5 3 soo.oo,,Tf ill O'cKson, uoraJ^y^ U0209 7348 Algebraic invanants' Mat 201 D5 lons^ liiiii^^ lllli ■!